Keywords: Mathematics Subject Classification (2010). 35J20, 35J70, 35P05, 35P30 Concave-convex problem, Nehari manifold, Dirichlet-Steklov boundary condition, elliptic problem, non-coerciveness, p-Laplacian, pbilaplacian, indefinite weights

On abstract indefinite concave-convex problems and applications to quasilinear elliptic equations

Mabel Cuesta and Liamidi Leadi

Abstract. In this work we study the existence of critical points of an abstract C 1 functional J defined in a reflexive Banach space X. This functional is of the form

J(u) = 1 p E(u) - 1 r A(u) - 1 q B(u),
with E, A, B positive-homogeneous indefinite functional of degree p, q, r respectively and 1 < p < q < r. The critical points are found by minimization along several subsets of the Nehari manifold associated to J.

We apply these results to various quasilinear elliptic problems, as for instance, the following p-laplacian concave-convex problem with Steklov boundary conditions on a bounded regular domain

     -∆pu + V (x)u p-1 = 0 in Ω;
|∇u| p-2 ∂u ∂ν = λa(x)u r-1 + b(x)u q-1 on ∂Ω; u > 0 in Ω, with given functions a, b, V possibly indefinite and 1 < r < p < q. We also apply our abstract result for a concave-convex quasilinear problem associated to the p-bilaplacian.

Introduction

After the celebrated paper of Ambrosetti-Brézis-Cerami [START_REF] Ambrosetti | Combined effect of concave and convex nonlinearities in some elliptic problems[END_REF] on the solvability of the elliptic problem

   -∆u = f λ (u)
in Ω, u = 0 on ∂Ω, u > 0 in Ω, for f λ (u) = λu r-1 + u q-1 and 1 < r < 2 < q, there have been a huge amount of research on this type of equations and the effects on multiplicity of the concave (1 < r < 2)-convex (2 < q) nonlinearity. In [START_REF] Brown | The Nehari manifold for a semilinear elliptic equation involving a sublinear term[END_REF][START_REF] Brown | A fibering map approach to a semilinear elliptic boundary value problem[END_REF][START_REF] Brown | The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function[END_REF] the authors studied the previous equation with a concave-convex forcing term f λ (u) = λa(x)u r-1 + b(x)u q-1 with changing-sign weights a(x), b(x) by using the so called Nehari manifold and the fibering map associated to the problem. This approach has proved to be very useful to deal with this type of problem and have become a subject of research on its own. The results of [START_REF] Ambrosetti | Combined effect of concave and convex nonlinearities in some elliptic problems[END_REF] were partially generalized to the p-laplacian operator under Dirichlet boundary conditions in [START_REF] Ambrosetti | Multiplicity results for some nonlinear elliptic equations[END_REF][START_REF] Garcia-Azorero | Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term[END_REF][START_REF] De Figueiredo | Local superlinearity and sublinearity for the p-Laplacian[END_REF][START_REF] Garcia-Azorero | Some results about the existence of a second positive solution in a quasilinear critical problem[END_REF][START_REF] Wu | Multiplicity of positive solution of p-Laplacian problems with signchanging weight functions[END_REF]. Soon later, other boundary conditions have been considered, as for instance in [START_REF] Sabina De Lis | A concave-convex quasilinear elliptic problem subject to a non linear boundary condition[END_REF]. Simultaneously some attention has been accorded to quasilinear problems that are non coercive. The first work in this direction was the study of the spectrum of the operator -∆ p u + V (x)|u| p-2 u with Dirichlet boundary conditions and V an indefinite bounded weight, see [START_REF] Cuesta | A weighted eigenvalue problem for the p-Laplacian plus a potential[END_REF]. In [START_REF] Ramos-Quoirin | Lack of coercivity in a concave-convex type equation[END_REF] the author studied the concave-convex problem    -∆ p u + V (x)u p-1 = λa(x)u r-1 + b(x)u q-1 in Ω; u = 0 on ∂Ω; u > 0 in Ω, by minimization on the Nehari set and obtained, under various "coerciveness" conditions related to V , a and b, the existence of up to four solutions: two solutions satisfying the condition E V (u) > 0 with

E V (u) := Ω (|∇u| p + V (x)|u| p ) dx,
and two more solutions satisfying E V (u) < 0.

Our goal in this work is to generalize the results of [START_REF] Ramos-Quoirin | Lack of coercivity in a concave-convex type equation[END_REF] for the same quasilinear operator, say -∆ p u + V (x)|u| p-2 u, but with different boundary conditions. Precisely, let Ω be a bounded smooth domain of class C 2,α (0 < α < 1) with outward unit normal ν on the boundary ∂Ω and ∆ p u := div (|∇u| p-2 ∇u) is the well known p-laplacian operator. The functions V ∈ L ∞ (Ω) and (a, b) ∈ (C s (∂Ω))

2 , for some s ∈ (0, 1), are allowed to change sign. The real number λ is a positive parameter. We will ask the exponents r, q to satisfy 1 < r < p < q < p * where p * = p(N -1) (N -p) + is the critical exponent for the trace operator W 1,p (Ω) → L s (∂Ω, dρ), and ρ denotes the restriction to ∂Ω of the (N -1)-Hausdorff measure, which coincides with the usual Lebesgue surface measure as ∂Ω is regular enough. We will consider the following quasilinear elliptic Problem I:

             -∆ p u + V (x)u p-1 = 0 in Ω;
|∇u| p-2 ∂u ∂ν = λa(x)u r-1 +b(x)u q-1 on ∂Ω; u > 0 in Ω, (1.1) and Problem II:

             -∆ p u + V (x)u p-1 = λa(x)u r-1
in Ω;

|∇u| p-2 ∂u ∂ν = b(x)u q-1 on ∂Ω; u > 0 in Ω.

(1.

2)

The search of solutions for these quasilinear problems can be stated in an abstract form as the search of critical points of a functional J defined on a Banach space X, which will take the form

J(u) = 1 p E(u) - 1 r A(u) - 1 q B(u), (1.3)
where E, A, B are possibly indefinite but positive-homogeneous functional of degree p, r, q respectively, with 1 < r < p < q. We will prove several existence and multiplicity results on critical points of J by minimizing J along several subsets of the Nehari manifold N = {u ∈ X \ {0}; J (u), u = 0} associated to J. These existence results for a general functional E, A, B and X can be applied in many cases, as for instance for the p-bilaplacian operator with Navier boundary conditions, that is the following Problem III:

∆ 2 p u -c|u| p-2 u = λa(x)|u| r-2 u + b(x)u q-2 u in Ω; u = ∆u = 0 on ∂Ω, (1.4)
with c ∈ IR. The p-bilaplacian operator is defined as ∆ 2 p u := ∆(|∆u| p-2 ∆u) and it has received recently some attention. In the case p = 2 the authors in [START_REF] Yang | On semilinear biharmonic equations with concaveconvex nonlinearities involving weight functions[END_REF] generalize the Ambrosetti-Brézis-Cerami problem in the case c = 0 (coercive case), indefinite weight a and non-negative weight b. See also [START_REF] Liu | Infinitely many solutions for p-biharmonic equation with general potential and concave-convex nonlinearity in IR N[END_REF] for similar results in IR N or [START_REF] Ji | On the p-biharmonic equation involving concave-convex nonlinearities and sign-changing weight function[END_REF] for Dirichlet boundary conditions (u = ∇u = 0) and p = 2. This paper is organized as follows. In section 2 we describe the Nehari set, the fibering map and the different "sign-subsets" of the Nehari set that will be used to find critical points of the C 1 functional J which is defined in (1.3). In sections 3 and 4 we prove four different critical points theorems of the functional J in the Nehari set, c.f. Theorem 3.3, Theorem 3.4, Theorem 4.1 and Theorem 4.2. The hypothesis needed to apply these general theorems concern the coerciveness of E along the sign-subsets of the Nehari set described in section 2. In section 5 we present various conditions in terms of eigenvalues-like numbers that will imply the required coerciveness and then we prove some existence results (see Theorem 5.1. and Theorem 5.7). In section 6 we apply the theorem of the previous section to state some existence and multiplicity results for problems I, II and III.

The Nehari set for a concave-convex functional

Let (X, • ) be a reflexive Banach space and E, A, B ∈ C 1 (X, IR). Let us assume that for some 1 < r < p < q it holds

E(tu) = t p E(u), A(tu) = t r A(u), B(tu) = t q B(u), ∀(t, u) ∈ IR + × X.
(2.1) The following hypothesis will also be assumed:

(H1) ∀(u n ) n∈IN , u n ∈ X, if u n
u for some u ∈ X then there exists a subsequence u n k such that

A(u n k ) → A(u) and B(u n k ) → B(u). (H2) E is bounded on bounded sets (i.e. sup u∈X, u =1 |E(u)| < ∞) and it is weakly lower semi-continuous. (H3) ∀(u n ) n∈IN , u n ∈ X, if u n u for some u ∈ X and E(u n ) → E(u) then u n → u.
By " " we denote the weak convergence in X. Let us consider the functional J defined as

J(u) = 1 p E(u) - 1 r A(u) - 1 q B(u)
and look for solutions of the problem J (u) = 0. Since J may be unbounded from below on the set {u ∈ X ; B(u) > 0}, it is useful to consider the functional J restricted to the so called Nehari set

N := {u ∈ X \ {0} ; N (u) := J (u), u = 0},
where •, • is the usual duality map defined on X × X. Thus u ∈ N if and only if u = 0 and

E(u) = A(u) + B(u).
Let us introduce the fibering maps associated to J. For u ∈ X \{0}, we define the function

J u : (0, ∞) -→ R t -→ J u (t) := J(tu).
Consequently u ∈ N if and only if J u (1) = 0 or yet, tu ∈ N (with t > 0) if and only if J u (t) = 0. The notation J u , J u stands here for

dJ u dt , d 2 J u dt 2 resp.
Furthermore, since J u (u) = 0 for u ∈ N we can write

J u (1) = (p -q)E(u) -(r -q)A(u) = (p -r)E(u) -(q -r)B(u). (2.2)
It is standard to split N into three sets that, roughly speaking, correspond to local minima, local maxima and inflexion points of J u :

N + := u ∈ X ; J u (1) = 0, J u (1) > 0 , N -:= u ∈ X ; J u (1) = 0, J u (1) < 0 , N 0 := u ∈ X ; J u (1) = 0, J u (1) = 0 .
We also introduce the following "sign-subsets":

A ± := {u ∈ X ; A(u) ≷ 0}, A 0 = {u ∈ X ; A(u) = 0}, B ± := {u ∈ X ; B(u) ≷ 0}, B 0 = {u ∈ X ; B(u) = 0}, E ± := {u ∈ X ; E(u) ≷ 0}, E 0 = {u ∈ X ; E(u) = 0}.
Finally we denote A ± 0 = A ± ∪ A 0 and similarly for B ± 0 and E ± 0 . We stress here that, without further assumptions on E, A, B, the previous sets can be void.

Next we state the following property of N that follows straight from hypothesis (H1)-(H3).

Lemma 2.1. Let u n be a sequence in N such that u n 0. Then u n → 0.

Let us give a result on the boundedness of N . We recall that a subset

C of X is called a cone if tx ∈ C, ∀(t, x) ∈ IR + × C. Lemma 2.2. Let C ⊂ X be weakly closed cone such that (C ∩ B 0 ) \ {0} ⊂ E + .
(2.3)

Then (i) the set N + ∩ C is bounded; (ii) if sup C∩N J(u) < ∞ then N ∩ C is bounded.
Proof. (i) Assume by contradiction that there is an unbounded sequence

u n ∈ N + ∩ C. Take v n = un un ∈ C and v 0 ∈ C such that v n v 0 . From the fact that u n ∈ N we have E(v n ) u n q-p = A(v n ) u n q-r + B(v n )
and passing to the limit we conclude that B(v 0 ) = 0. We have used here that E is bounded on bounded sets (i.e. (H2)). From the fact that u n ∈ N + we have

E(v n ) < q -r q -p A(v n ) u n p-r
and passing to the limit it comes that E(v 0 ) ≤ 0. The possibility of v 0 = 0 is ruled out by the fact that, in that case, 0 = E(v 0 ) = lim inf n→∞ E(v n ) and therefore, by (H3), we will have v n → v 0 = 0. This is a contradiction with the property v n = 1. Thus v 0 = 0 and we get a contradiction with the hypothesis (2.3) of the lemma.

(ii) Assume that u n ∈ N ∩ C is a sequence satisfying u n → +∞ and denote v n = un un . Since the sequence v n is bounded, there exists v 0 ∈ X and a subsequence v n k such that v n k v 0 . From the fact that u n k ∈ N we have, as previously, 0 = B(v 0 ). On the other hand

J(u n k ) u n k p = 1 p - 1 q E(v n k ) - 1 r - 1 q A(v n k ) u n k p-r
therefore passing to the limit we get

E(v 0 ) ≤ lim inf k→+∞ E(v n k ) ≤ 0 because J(u n ) is uniformly bounded from above. If v 0 = 0 hence E(v 0 ) = 0 = lim inf k→+∞ E(v n k )
and, from hypothesis (H3), v n k → v 0 = 0. This is impossible because v n k = 1 for all k. Thus v 0 = 0 and we get again a contradiction with the hypothesis (2.3) of the lemma.

We also have the following property:

Lemma 2.3. Let C ⊂ X be weakly closed cone and assume that

C ∩ A 0 \ {0} ⊂ E + . (2.4)
Then N -∩ C has no sequence u n 0.

Proof. Assume by contradiction that there is a sequence u n ∈ N -∩ C such that u n 0 in X. From the fact that u n ∈ N we have 0 = E(0) ≤ lim inf E(u n ) = lim inf A(u n ) + B(u n ) so u n → 0 on X. Take z n := un un ∈ C and assume that for some z 0 ∈ X we have z n z 0 , A(z n ) → A(z 0 ) and B(z n ) → B(z 0 ). By using that u n ∈ N we have

A(z n ) = E(z n ) u n p-r -B(z n ) u n q-r
and passing to the limit it comes A(z 0 ) = 0. Besides by using that u n ∈ N - and (2.2) we have

E(z n ) ≤ q -r p -r B(z n ) u n q-p ,
and passing to the limit

E(z 0 ) ≤ lim inf E(z n ) ≤ 0.
Notice that the possibility z 0 = 0 is excluded because, in that case, we would have 0 = E(z 0 ) = lim inf E(z n ) which will imply that z n → z 0 = 0, a contradiction with the fact that z n = 1. Thus we have proved that z 0 ∈ C ∩ A 0 \ {0}. Then from the hypothesis (2.4) of this lemma it comes that E(z 0 ) > 0, a contradiction.

The fibering map

Let us give a complete description of the behaviour of J u according to the sign of A(u), B(u) and E(u). Let us write

J u (t) = t p-1 E(u) -t r-1 A(u) -t q-1 B(u) = t r-1 [m u (t) -A(u)],
where m u (t) := t p-r E(u) -t q-r B(u).

(2.5) Clearly, for t > 0, tu ∈ N if and only if t is a solution of the equation

m u (t) = A(u).
(2.6)

If for u ∈ X\{0} and t > 0 one has that J u (t) = 0 then J u (t) = t r-1 m u (t).

Consequently tu ∈ N + if and only if m u (t) > 0 (similar results for N -and N 0 ). In order to study the resolvability of (2.6) let us describe the variation of the function t → m u (t) for any u ∈ E 0 ∩ B 0 . Four possible pictures of the graph of m u can be drawn: We shall now describe the nature of the fibering maps for all possible signs of A(u), B(u) and E(u). The following behaviour of the function J u follows from the previous description of the function m u . As above, we will assume that u ∈ E 0 ∩ B 0 . Case 1 : E(u) > 0 and B(u) > 0.

In this case the function m u has graph as shown in Figure 1(a).

Case 1.1 : If A(u) > 0 and A(u) < max t>0 m u (t) then it is clear that there are exactly two solutions 0 < t 1 (u) < t 2 (u) of (2.6) with m u (t 2 (u)) < 0 < m u (t 1 (u)). Thus there are exactly two multiples of u lying in N , namely t 1 (u)u ∈ N + and t 2 (u)u ∈ N -. It follows that J u has exactly two critical points, a local minimum at t 1 (u) and a local maximum at t 2 (u). Moreover J u is decreasing in (0, t 1 (u)) and in (t 2 (u), ∞), increasing in (t 1 , t 2 ), see Figure 2(f). Case 1.2 : If A(u) ≤ 0. Then using the graph of m u in Figure 1(a), we deduce that there exists one positive solution of (2.6). Consequently J u has graph as shown in Figure 2(g) and there is a unique value t(u) > 0 such that t(u)u ∈ N . Moreover m u (t(u)) < 0, so t(u)u ∈ N -and the fibering maps J u has a unique critical point which is a local maximum. Case 2 : E(u) ≥ 0 and B(u) ≤ 0.

In this case the function m u is an increasing function of t (see Figure 1(b)). Case 2.1 : If A(u) > 0, then m u has a graph as in Figure 1(b) and J u has a graph as shown in Figure 2(i). It is clear that there is exactly one solution of (2.6), i.e. there is a unique t(u) > 0 such that t(u)u ∈ N . Moreover m u (t(u)) > 0 (since m u is an increasing function of t) and so t(u)u ∈ N + . Thus the fibering map J u has a unique critical point which is a local minimum, as shown in Figure 2(i). Case 2.2 : If A(u) ≤ 0 then the function J u is increasing functions of t and so has graph as shown in Figure 2(h). Consequently (2.6) has no solution, for all t and thus no multiple of u lies in N . Case 3 : E(u) ≤ 0 and B(u) ≥ 0.

In this case the function m u is a decreasing function of t and has graph as shown in Figure 1(c). Case 3.1 : If A(u) < 0 then (2.6) has a unique solution. Since J u must have graph as shown in Figure 2(g), we conclude again that there is a unique t(u) > 0 such that t(u)u ∈ N and since m u (t(u)) < 0 in this case, we deduce that t(u)u ∈ N -. Hence the fibering map J u has a unique critical point which is a local maximum. Case 3.2 : If A(u) ≥ 0 then (2.6) has no solution. Moreover J u is a decreasing function of t and has graph as shown in Figure 2(e). Thus in this case no multiple of u lies in N . Case 4 : E(u) < 0 and B(u) < 0.

In this case m u has graph as shown in Figure 1(d). Case 4.1 : If A(u) < 0 and A(u) > min t>0 m u (t) then J u has graph as shown in Figure 2(j). In this case there are exactly two solutions t 1 (u) < t 2 (u) of (2.6) with m u (t 1 (u)) < 0 < m u (t 2 (u)). Thus there are exactly two multiples of u which belong to N , namely t 1 (u) ∈ N - and t 2 (u) ∈ N + . It follows that J u has exactly two critical points, a local maximum at t = t 1 (u) and a local minimum at t = t 2 (u). Furthermore J u is increasing in (0, t 1 ), decreasing in (t 1 , t 2 ) and increasing in (t 2 , ∞), as in Figure 2(j). Case 4.2 : If A(u) ≥ 0 then (2.6) has a unique solution and J u has graph as shown in Figure 2(i). Thus there is a unique value t(u) > 0 such that t(u)u ∈ N . Since m u (t(u)) > 0, we deduce that t(u)u ∈ N + and consequently J u has a unique critical point which is a local minimum as shown in Figure 2(i).

(i) If either A + ∩ B - 0 ∩ E + = ∅ or Λ + := {u ∈ A + ∩ B + ∩ E + ; A(u) < max t>0 m u (t)} = ∅ then N + ∩ E + = ∅. (ii) If either A - 0 ∩ B + ∩ E + = ∅ or Λ + = ∅ then N -∩ E + = ∅. (iii) If Λ -:= {u ∈ A -∩ B -∩ E -; A(u) > min t>0 m u (t)} = ∅ then N + ∩ A -= ∅. (iv) If Λ -= ∅ then N -∩ B -= ∅.
A simple calculation shows that the maximum of m u in case I is

max t>0 m u (t) = q -p q -r p -r q -r p-r q-p E(u) q-r q-p B(u) p-r q-p
and the minimum of m u in case IV is

min t>0 m u (t) = - q -p q -r p -r q -r p-r q-p (-E(u)) q-r q-p (-B(u)) p-r q-p .
The maximum (resp. the minimum) of m u is achieved at the point

t * (u) := (p -r)E(u) (q -r)B(u) 1 q-p .
(2.7)

Let us denote

λ + * := inf u∈A + ∩B + ∩E + max t>0 m u (t) A(u) ≥ 0 (2.8)
in case I and

λ - * := inf u∈A -∩B -∩E - min t>0 m u (t) A(u) ≥ 0 (2.9)
in case IV. From the previous discussion it follows trivially that:

Lemma 2.5. (i) If λ + * > 1 then N 0 ∩ E + = ∅. (ii) If λ - * > 1 then N 0 ∩ E -= ∅.
3. Local minimizers of J restricted to the Nehari set and to E + Our purpose in this section is to prove that, under some suitable assumptions on A ± , B ± and E ± , the functional J is bounded below and achieves its infimum on some of the sign subsets of N described in cases I to IV of section 2. This will provide us critical points for J, as a consequence of the following well known result:

Lemma 3.1. Suppose that u is a local minimiser of J restricted to N . If u ∈ N 0 then u is a critical point of J relative to X.
Proof. Since u is a minimiser of J on N , there exists γ ∈ R (Lagrange multiplier) such that

J (u) = γN (u). (3.1)
Thus in particular we have

J (u), u = γ N (u), u , which implies that γ N (u), u = 0 because 0 = N (u) = J (u), u (since u ∈ N ). Moreover N (u), u = pE(u) -rA(u) -qB(u) = (p -r)E(u) -(q -r)B(u) = J u (1).
Consequently, if u ∈ N 0 , that is J u (1) = 0,then γ = 0 and we conclude from (3.1) that u is a critical point of J.

Let us rewrite the functional J for u ∈ N in two different forms:

J(u) = 1 p - 1 r E(u) + 1 r - 1 q B(u) = 1 p - 1 q E(u) - 1 r - 1 q A(u). (3.2)
We then observe the following

Lemma 3.2. (a) J(u) > 0 for all u ∈ (N -∩ A - 0 ) ∪ (N -∩ B + 0 ) ∪ (N -∩ E - 0 ); (b) J(u) < 0 for all u ∈ (N + ∩ A + 0 ) ∪ (N + ∩ B - 0 ) ∪ (N + ∩ E + 0 ). Proof. From (2.2) we deduce the following inequalities (a) If u ∈ N -then q -r q -p A(u) < E(u) < q -r p -r B(u). (3.3) Hence J(u) > max (q -r)(r -p) prq A(u), (q -p)(r -p) pqr E(u), (q -p)(q -r) pqr B(u) , (b) Similarly, if u ∈ N + then q -r p -r B(u) < E(u) < q -r q -p A(u). (3.4)
Hence J(u) < min (q -r)(r -p) pqr A(u), (q -p)(r -p) pqr E(u), (q -p)(q -r) pqr B(u) .

Minimizing J along N +

A first critical point of J can be found on N + ∩ E + , provided this set is not empty and some "coerciveness" conditions:

Theorem 3.3. Assume that E + ∩ N + = ∅, λ + * > 1 and (H4) (A + 0 ∩ B 0 ) \ {0} ⊂ E + , (H5) A + ⊂ E + . Then the following local infimum i := inf u∈N + ∩E + J(u)
is achieved. Furthermore i < 0.

Proof. The fact that i < 0 readily follows from Lemma 3.2(b). First we prove that N + ∩ E + is bounded. Indeed N + ∩ E + ⊂ N + ∩ A + because of inequality (3.4) and, since A + 0 is a weakly closed cone, we have that the condition (2.3) of Lemma 2.2 corresponds to hypothesis (H4), thus we conclude that N + ∩E + is bounded. Notice that in particular we have that i > -∞. To prove that the infimum i is achieved take a minimizing sequence u n . Since this sequence is bounded there exists some u 0 ∈ X such that, up to subsequence, u n u 0 . By using (3.2) we can write

J(u n ) = 1 p - 1 q E(u n ) - 1 r - 1 q A(u n ) ≥ - 1 r - 1 q A(u n ).
and letting n → +∞ we get A(u 0 ) ≥ -i( 1 r -1 q ) -1 > 0. Then it comes then from (H5) that E(u 0 ) > 0. We claim that u n converges strongly to u 0 in X. Assume by contradiction that u n → u 0 . We discuss two alternatives : Alt. 1. B(u 0 ) > 0. Using the previous classification of the fibering maps, the graph of J u0 is as the one in Figure 2(f) so there exist 0 < t 1 (u 0 ) < t 2 (u 0 ) such that t 1 (u 0 )u 0 ∈ N + ∩E + , t 2 (u 0 )u 0 ∈ N -∩E + , J u0 is increasing between t 1 (u 0 ) and t 2 (u 0 ) and decreasing elsewhere. Since we are assuming that u n → u 0 hence u 0 ∈ N and therefore t 1 (u 0 ) = 1. Let us distinguish two cases: (a) 1 ≤ t 2 (u 0 ) and (b) 1 > t 2 (u 0 ) . In case (a) we have

J(t 1 (u 0 )u 0 ) = J u0 (t 1 (u 0 )) ≤ J u0 (1) < lim inf J un (1) = lim J(u n ) = i. (3.5)
Moreover Lemma 2.5 and hypothesis (H4) imply that N 0 = {0}. Thus (3.5) leads to a contradiction because t 1 (u 0 )u 0 ∈ N + ∩ E + . In case (b) using that

0 = J u0 (t 2 (u 0 )) < lim inf J un (t 2 (u 0 ))
we conclude that J un (t 2 (u 0 )) > 0 for n large. Since 1 is a local minimum of J un and the graph of J un looks like the one of figure 2(f) then it must be 1 < t 2 (u 0 ), a contradiction. Alt. 2. B(u 0 ) ≤ 0. The graph of J u is as the one in Figure 2(i) so there exists 0 < t(u 0 ) such that t(u 0 )u 0 ∈ N + ∩ E + and J u0 has a global minimum in t(u 0 ). If t(u 0 ) ≥ 1 again we have (3.5), a contradiction. If t(u 0 ) < 1 we use that 0 = J u0 (t(u 0 )) < lim inf J un (t(u 0 )), so then J un (1) > 0 for n large, which is also impossible because u n ∈ N .

Minimizing J along N -

We now look for solutions of

J (u) = 0 in N -∩ E + . Theorem 3.4. Let us assume that N -∩ E + = ∅, λ + * > 1 and (H6) B + 0 \ {0} ⊂ E + , Then the following infimum j := inf u∈N -∩E + J(u) is > 0 and it is achieved.
Proof. We know from Lemma 3.2(a) that j ≥ 0. Observe that any minimizing sequence is bounded because (H6) implies the hypothesis (2.3) of Lemma 2.2 and the result comes from (ii) of the aforementioned lemma. Assume first that j > 0. We claim that there is a strong convergent minimizing sequence for j. Assume by contradiction that we have a minimizing sequence u n u 0 such that A(u n ) → A(u 0 ), B(u n ) → B(u 0 ) but u n → u 0 . If u 0 = 0 then Lemma 2.1 will imply that u n → 0, which is not the case we are assuming now. Thus u 0 = 0. We can also prove that B(u 0 ) > 0 by using the fact that

u n ∈ E + and q -r qr B(u n ) = J(u n ) - 1 p - 1 r E(u u ) ≥ J(u n ).
Then passing to the limit we get B(u 0 ) > 0. Consequently, (H6) implies that E(u 0 ) > 0. Now we distinguish two cases according to the sign of A(u). Alt. 1. A(u 0 ) > 0. In this case J u0 and J un look like Figure 2(f). If u n → u 0 then t 2 (u 0 ) = 1. We also have that t 2 (u 0 )u 0 ∈ N -∩ E + (we use here that λ + * > 1 to assure that t 2 (u 0 )u 0 ∈ N 0 , c.f. Lemma 2.5). Furthermore 0 = J u0 (t 2 (u 0 )) < lim inf J un (t 2 (u 0 )).

Thus J un (t 2 (u 0 )) > 0 for n large. Since t 2 (u n ) = 1 hence t 1 (u n ) < t 2 (u 0 ) < 1 and we will have

j ≤ J u0 (t 2 (u 0 )) < lim inf J un (t 2 (u 0 )) ≤ lim n→+∞ J un (1) = j, (3.6)
a contradiction. Alt. 2. A(u 0 ) ≤ 0. In this case J u0 and J un look like Figure 2(g). If u n → u 0 then t(u 0 ) = 1. We also have that t(u 0 )u 0 ∈ N -∩ E + . Then again we have

j ≤ J u0 (t(u 0 )) < lim inf J un (t(u 0 )) ≤ lim n→+∞ J un (1) = j, (3.7)
a contradiction.

Let us finally prove that j > 0. If this were not the case then for any minimizing sequence (which we know that will be bounded) we will have J(u n ) → 0. Then, up to a subsequence, there exists u 0 ∈ X such that u n u 0 , A(u n ) → A(u 0 ) and B(u n ) → B(u 0 ). From Lemma 3.2(a) we know that N -∩ E + ⊂ B + and hence the possibility u 0 = 0 is excluded from Lemma 2.3 and (H6) applied to C = B + 0 , so (2.4) is satisfied. Besides by writing

0 = lim n→+∞ J(u n ) = lim n→+∞ 1 p - 1 r E(u n ) + 1 r - 1 q B(u n ) it comes lim n→+∞ E(u n ) = p(q -r) q(p -r) B(u 0 ). (3.8) Thus, if B(u 0 ) = 0 then E(u 0 ) ≤ lim n→+∞ E(u n ) = 0, a contradiction with (H6). Hence B(u 0 ) > 0. Furthermore, from (3.8) and 0 = lim n→+∞ J(u n ) = lim n→+∞ 1 p - 1 q E(u n ) - 1 r - 1 q A(u n ) it comes A(u 0 ) = r(q -p) q(p -r) B(u 0 ) > 0.
We are going to rule out the following two alternatives: Alt. 1. u n → u 0 . In this case J(u 0 ) = 0 and we will have u 0 ∈ N -∩ E + (the possibility of u 0 ∈ N 0 is ruled by the constraint λ + * > 1). But, according to Lemma 3.2(a), J(u 0 ) > 0, a contradiction. Alt. 2. u n → u 0 . The maps J u0 will look as in Figure 2(f) and repeating the argument of Alt. 1 above we have (3.6) and we reach a contradiction. This conclude the proof of j > 0. We first look for solutions of our problem in

N + ∩ A -⊂ N + ∩ E -. Theorem 4.1. Assume that N + ∩ A -= ∅, λ - * > 1 and (H7) A - 0 ∩ B 0 \ {0} ⊂ E + (H8) A 0 ∩ B -⊂ E + .
Then the following infimum

l := inf u∈N + ∩A -J(u)
is < 0 and it is achieved.

Proof. It comes from (H7) and Lemma 2.2(i) that N + ∩ A -is bounded and by Lemma 3.2(a) that l is negative. Let us prove that this infimum is achieved. Let u n be a minimizing sequence. Since this sequence is bounded, there exists

u 0 such that u n u 0 , A(u n ) → A(u 0 ) and B(u n ) → B(u 0 ). Thus A(u 0 ) ≤ 0, B(u 0 ) ≤ 0 and E(u 0 ) ≤ lim inf E(u n ) ≤ 0. From E(u n ) ≤ 0 and J(u n ) = 1 p - 1 r E(u n ) + 1 r - 1 q B(u n )
we infer that

B(u n ) ≤ rq q -r J(u n )
and passing to the limit it comes that B(u 0 ) < 0. In particular u 0 = 0. Now, if A(u 0 ) = 0, using (H8) we will conclude that E(u 0 ) > 0, a contradiction. Then A(u 0 ) < 0 and J u0 behaves either as in Figure 2(h) if E(u 0 ) = 0 or as in Figure 2(j) if E(u 0 ) < 0. Let us discuss this two alternatives: Alt. 1. E(u 0 ) = 0. In this case E(u 0 ) = 0 = lim inf E(u n ) = 0 so u n → u 0 by ( H3). Hence J(u 0 ) = l and also u 0 ∈ N + 0 ∩ A -. Since be in N 0 implies that E(u 0 ) = q-r q-p A(u 0 ) by (3.4) and we know that A(u 0 ) < 0 = E(u 0 ), then clearly u 0 ∈ N 0 , so have u 0 ∈ N + ∩ A -and we are done. Alt. 2. E(u 0 ) < 0. In this case there exist two values 0 ≤ t 1 (u 0 ) < t 2 (u 0 ) such that t 2 (u 0 ) > 0 is a global minimum value for J u0 , t 2 (u 0 )u 0 ∈ N + ∩ A - and J u0 is decreasing between t 1 (u 0 ) and t 2 (u 0 ), increasing elsewhere. Notice that u 0 ∈ N 0 because of condition λ - * > 1. Let us assume by contradiction that u n → u 0 . Then

l ≤ J(t 2 (u 0 )u 0 ) = J u0 (t 2 (u 0 )) ≤ J u0 (1) < lim inf J un (1) = lim J(u n ) = l, a contradiction. 4.2. Minimizing J along N - Finally we minimize along N -∩ B -⊂ N -∩ E -. Theorem 4.2. Assume that N -∩ B -= ∅, λ - * > 1, (H7) and (H9) A 0 ∩ B - 0 \ {0} ⊂ E + . Then the infimum k := inf u∈N -∩B -J(u)
is achieved and it is positive.

Proof. By Lemma 3.2(a) the value k ≥ 0. Since N -∩ B -⊂ A -, Lemma 2.2(ii) and (H7) imply that any minimizing sequence is bounded. We claim that any minimizing sequence u n possesses a convergence subsequence. Indeed, assume

u n u 0 , A(u n ) → A(u 0 ) ≤ 0 and B(u n ) → B(u 0 ) ≤ 0 and that u n → u 0 . Since u n ∈ N -we have E(u n ) ≤ q-r p-r B(u n ) ≤ 0 so E(u 0 ) ≤ 0.
Let us assume first that k = 0. If u 0 = 0 then from Lemma 2.1 it comes that u n → 0 and therefore 0 = lim J(u n ) = k, a contradiction. Thus u 0 = 0. From hypothesis (H7) and (H9) we must have u 0 ∈ A -∩ B -. Hence J u0 looks like Figure 2(j) if E(u 0 ) < 0 or like Figure 2(h) if E(u 0 ) = 0. Let us to discuss this two alternatives. Alt. 1. E(u 0 ) = 0. As we have E(u 0 ) = 0 = lim inf E(u n ) then u n → u 0 . Hence, using that u n ∈ N and passing to the limit we find 0 = E(u 0 ) = A(u 0 ) + B(u 0 ) ≤ 0 so it must be A(u 0 ) = B(u 0 ) = 0, a contradiction with (H7) or (H9). We have ruled out this alternative. Alt. 2. E(u 0 ) < 0. Thus there exists 0 < t 1 (u 0 ) < t 2 (u 0 ) such that t 1 (u 0 )u 0 ∈ N -and t 1 (u n ) = 1. We have used here that t 1 (u 0 )u 0 ∈ N 0 because of the hypothesis λ - * > 1. Let us assume by contradiction that u n → u 0 . Hence 0 = J u0 (t 1 (u 0 )) < lim inf J un (t 1 (u 0 )) which implies that J un (t 1 (u 0 )) > 0 for n large. Thus t 1 (u 0 ) < 1 or t 1 (u 0 ) > t 2 (u n ) because J un looks line Figure 2(j). In the first case we have

k ≤ J(t 1 (u 0 )u 0 ) = J u0 (t 1 (u 0 )) < lim inf J un (t 1 (u 0 )) < lim inf J un (1) = k,
a contradiction. In the second case let us suppose, up to a subsequence, that t 2 (u n ) converges to some s ∈ [1, t 1 (u 0 )]. We have

J u0 (s)) < lim inf J un (t 2 (u n )) = 0,
which is a contradiction because J u0 > 0 on (0, t 1 (u 0 )). We have just proved that the minimizing sequence converges strongly and consequently the infimum k is achieved.

To finish the proof let us check that k > 0. Assume by contradiction that k = 0 and take u n a minimizing sequence. Since u n is bounded (as above) then we can assume that u n u 0 for some u 0 . We have B(u 0 ) ≤ 0, A(u 0 ) ≤ 0 and u 0 ∈ E - 0 . It follows from Lemma 2.3 and (H9) that u 0 = 0 and from the fact that u n ∈ N -we have

A(u n ) > pqr (q -r)(r -p) J(u n ).
Hence passing to the limit we get A(u 0 ) = 0. Thus u 0 ∈ A 0 ∩ B - 0 ∩ E - 0 which contradicts (H9) if B(u 0 ) = 0 or (H7) if B(u 0 ) = 0.

On various conditions for coerciveness and existence results

5.1. On the coerciveness of E restricted to A ± 0 and B ± 0

We want to give in this section some variational conditions on E, A, B that would imply hypothesis (H4)-(H9). Those conditions will concern the following constants of coerciveness

i ± (A) := inf {E(u) ; A(u) = ±1} , i ± (B) =: inf {E(u) ; B(u) = ±1} , (5.1)
j 0 (A) := inf {E(u) ; A(u) = 0, u ∈ S} , j 0 (B) := inf {E(u) ; B(u) = 0, u ∈ S} . (5.2)
Recall that S is the unit sphere of X. We give below and existence and multiplicity result for the equation J (u) = 0 in terms of the constants j 0 , i ± .

Theorem 5.1. Let us assume hypothesis (H1) to (H3).

(i) If λ + * > 1, N + ∩ E + = ∅, i + (A)
> 0 and either j 0 (A) > 0 or j 0 (B) > 0 then there exists at least one solution of

J (u) = 0 in N + ∩ E + . (ii) If λ + * > 1, N -∩ E + = ∅, j 0 (B) > 0 and i + (B) > 0 then there exists at least one solution of J (u) = 0 in N -∩ E + . (iii) If λ - * > 1, N + ∩ A -= ∅
and either (1) j 0 (A) > 0 and i -(A) > 0 or (2) j 0 (B) > 0 and j 0 (A) > 0 or (3) j 0 (B) > 0 and i -(B) > 0, then there exists at least one solution of

J (u) = 0 in N + ∩ A -⊂ N + ∩ E -. (iv) If λ - * > 1, N -∩ B -= ∅
and either (4) j 0 (A) > 0 and j 0 (B) > 0 or (5) j 0 (A) > 0 and i -(A) > 0 or (6) j 0 (B) > 0 and i -(B) > 0, then there exists at least one solution of

J (u) = 0 in N -∩ B -⊂ N -∩ E -.
We left the proof to the reader. In the next proposition we give a variational characterization of i ± (A) and i ± (B). Proposition 5.2. Let us assume hypothesis (H1) to (H3). Assume that j 0 (A) > 0 and that i ± (A) ∈ IR. Then there exits ϕ

± ∈ X satisfying A(ϕ ± ) = ±1 such that E (ϕ ± ) = p r i ± (A)A (ϕ ± ).
Similarly, if j 0 (B) > 0 and i ± (B) ∈ IR then there exits φ ± ∈ X satisfying B(φ ± ) = ±1 such that

E (φ ± ) = p q i ± (B)B (φ ± ).
Proof. We only prove the first part. If u n ∈ X with A(u n ) = 1 is a minimizing sequence for, say, i + (A) then the sequence u n is bounded. Indeed, otherwise if we take v n := vn un hence, up to a subsequence, there exists v 0 ∈ X such that v n v 0 , A(v n ) → 0 and E(v n ) → 0. Notice that v 0 = 0 because otherwise it will follows from 0 = E(v 0 ) ≤ lim inf E(v n ) = 0 and (H3) that v n → 0. This is impossible since v n = 1. We then have v 0 = 0, A(v 0 ) = 0 and E(v 0 ) ≤ lim inf E(v n ) = 0, a contradiction with the hypothesis j 0 (A) > 0. We have just proved that u n is bounded. The proof that i + (A) is achieved is standard and we omit it here. Let us denote by ϕ + an element of X where the infimum is achieved. From the Lagrange multipliers rule it follows that there exists λ ∈ IR such that E (ϕ + ) = λA (ϕ + ). By testing this last equation against ϕ + and using the different homogeneities of E and A we get the result.

Remark 5.3. As a matter of fact the constraint A(u) = 1 in the definition of i + (A) := i + (A, 1) can be replaced by, say, A(u) = c, where c in any positive number. In that case, it is trivial to prove that i + (A, c) = c p r i + (A, 1). A similar statement can be formulated for i -(A) and i ± (B).

Remark 5.4. We can easily prove that j 0 (A) and j 0 (B) are achieved provided they are finite and j 0 (A, B) := inf{E(u) ; A(u) = 0, B(u) = 0, u ∈ S} > 0. However we can not give a variational formulation of any of them mainly because, in general, we don't know if the set A -1 ({0}) (resp. B -1 ({0})) is a manifold.

On the coerciveness of E and a principal eigenvalue

In this section we look for sufficient conditions implying (H4) to (H9) in terms of the first eigenvalue of the operator E. Precisely, let us assume the following

(H3) ∃(Y, • Y )a Banach space s.tX is cont. one-to-one embeded onY, X → Y is compact and u → u p Y is Fréchet differentiable. Let us denote χ(u) = u p Y . Then it is straightforward from (H2)-(H3)' that λ 1 := inf{E(u) ; u Y = 1} (5.3) is > -∞ and it is achieved. Moreover, if E(ϕ) = λ 1 and ϕ Y = 1 then, by Lagrange multiplier rule, E (ϕ) = λ 1 χ (ϕ).
We can give now the following sufficient conditions in terms of λ 1 to assure (H4)-(H9). Notice that the following conditions are stronger than the ones in terms of i ± (A), i ± (B) etc. Let us also stress here that λ 1 > 0 is equivalent to E - 0 \ {0} = ∅ and therefore (H4) to (H9) are trivially satisfied. Hereafter we denote

E λ1 := {ϕ ∈ X \ {0} ; E (ϕ) = λ 1 χ (ϕ)}.
We have the sequence v n = un un will satisfy E(v n ) → 0, v n Y → 0, so, up to a subsequence, v n 0, in contradiction with (H3) and the fact that v n = 1). Let u 0 ∈ X be such that u n u 0 . Thus A(u 0 ) ≥ 0, u 0 Y = 1 and consequently d ≤ E(u 0 ) ≤ lim inf E(u n ) = d. We have proved that E(u 0 ) = d and d is achieved. If A(u 0 ) = 0 hence d ≥ j 0 (A) > 0 and we are done. If A(u 0 ) > 0 hence

d = inf{E(u) ; A(u) > 0, u Y = 1}
and d is then a local minima of E under the constrain u Y = 1. By Lagrange multiplier rule, there exists λ ∈ IR such that E (u 0 ) = λχ (u 0 ). By evaluating this equation at u 0 we readily obtain

pd = pE(u 0 ) = E (u 0 ), u 0 = λ χ (u 0 ), u 0 = pλ.
Since E λ1 ⊂ A -then λ = λ 1 and therefore d = λ > 0 by (H3) .

We can now give the first main existence result of this section. This result generalizes Theorem 4.8 and Theorem 5.7 of [START_REF] Ramos-Quoirin | Lack of coercivity in a concave-convex type equation[END_REF]. Theorem 5.7. Let us assume hypotheses (H1) to (H3) .

(i) Assume that N ± ∩ E + = ∅ and λ + * > 1. If either λ 1 > 0 or λ 1 = 0 and E λ1 ⊂ A -∩ B -then there exists at least two solutions of J (u) = 0. (ii) Assume that N + ∩ E + = ∅ and λ + * > 1. If λ 1 = 0 and E λ1 ⊂ A -∩ B + 0
then there exists at least one solution of J (u) = 0. (iii) Assume that N + ∩ A -= ∅, N -∩ B -= ∅ and λ - * > 1. If j 0 (A) > 0 and j 0 (B) > 0 then there exist at least two solutions of J (u) = 0 in E -. (iv) Assume (H3) , N + ∩ E + = ∅, λ + * > 1 and λ 1 < 0. If j 0 (A) > 0 and E λ1 ⊂ A -then there exist at least one solution of

J (u) = 0 in E + . (v) Assume (H3) , N ± ∩ E + = ∅, λ + * > 1 and λ 1 < 0. If j 0 (A) > 0, j 0 (B) > 0 and E λ1 ⊂ A -∩ B -then
(i) λ 1 > 0 ⇒ j ± 0 (A, B) > 0, (ii) If λ 1 = 0 then (1) E λ1 ∩ A + 0 ∩ B + 0 = ∅ ⇒ j + 0 (A, B) > 0. (2) E λ1 ∩ A - 0 ∩ B - 0 = ∅ ⇒ j - 0 (A, B) > 0.
Let us show that, in a particular case, the variational equation (5.6) is equivalent to the equation J (u) = 0. By equivalent we mean here that a multiple of a solution of (5.6) is a solution of J (u). Let us observe that if ϕ ∈ X \ {0} is a solution of (5.6) and we denote for simplicity

d 1 := (β/α) α (λ + * ) β A(ϕ) , d 2 := (α/β) β (λ + * ) β B(ϕ) (5.7)
then for any c > 0 the function v = cϕ satisfies

E (v) = d 1 c p-r A (v) + d 2 c p-q B (v).
Thus we have the following result Proposition 5.10. Assume that j + 0 (A, B) > 0 and that

(λ + * ) β = p r β . p q α . (5.8)
Then there exists a solution of J (u) = 0 satisfying u ∈ A + ∩B + . The constant c pqr := ( p r ) β .( p q ) α is > 1. A similar result can be state for λ + * if j - 0 (A, B) > 0. Proof. Recall that J (u) = 0 if and only if E (u) = p r A (u) + p q B (u). Let ϕ ∈ X be a point where λ * + is achieved. A simple computation shows that we can choose c > 0 such that cϕ is a critical point of J, that is, d 1 c p-r = p r and d 2 c p-q = p q if and only if

p r d 1 β . p q d 2 α = 1
and using the fact that α + β = 1, A(ϕ) β B(ϕ) α = 1 and (5.7) we get (5.8).

Let us write

(c pqr ) q-r r = p r q r -1 . q r 1-p r .
One has q -r r ln(c pqr ) = ( q r -1) ln p r + (1 -p r ) ln q r = (x -1) ln y -(y -1) ln x, where we have put x := q r and y := p r . Using the fact that the function f (z) := ln z z-1 is strictly decreasing for z > 1 we conclude that q-r r . ln(c pqr ) > 0 and therefore the constant c pqr > 1 as claimed.

Remark 5.11. Notice that, if (5.8) holds, we can not distinguish the solution of the problem J (u) = 0 obtained in Proposition 5.10 from the one obtained in Theorem 3.3 or the one in Theorem 3.4.

Existence results for quasilinear problems 6.1. Existence and multiplicity results for Problem I

Let Ω ⊂ IR N be a bounded smooth domain of class C 2,α (0 < α < 1) with outward unit normal ν on the boundary ∂Ω. Let V ∈ L ∞ (Ω) and (a, b) ∈ (C s (∂Ω)) 2 , for some s ∈ (0, 1), and allow them to change sign. The exponents r, q to satisfy 1 < r < p < q < p * where p * = p(N -1) (N -p) + the critical exponent for the trace embedding; ρ denotes the restriction to ∂Ω of the (N -1)-Hausdorff measure, which coincides with the usual Lebesgue surface measure as ∂Ω is regular enough. Finally let the number λ be a positive parameter.

Take X = W 1,p (Ω) with the usual Sobolev norm • 1,p . Solutions of Problem I are understood in the weak sense, that is,

Ω |∇u| p-2 ∇u∇ϕ+V (x)|u| p-2 uϕ = ∂Ω λa|u| r-2 u+b|u| q-2 u ϕ dρ. (6.1) ∀ϕ ∈ W 1,p (Ω). Let us consider E(u) = Ω (|∇u| p + V (x)|u| p ) dx, A(u) = λ ∂Ω a(ρ)|u| r dρ, B(u) = ∂Ω b(ρ)|u| q dρ,
and the energy functional

J(u) = 1 p E(u) - 1 r A(u) - 1 q B(u) = = 1 p Ω (|∇u| p + V (x)|u| p ) dx - λ r ∂Ω a(ρ)|u| r dρ - 1 q ∂Ω b(ρ)|u| q dρ.
It is clear that solutions of Problem I are positive critical points of J. It is also clear that A, B satisfy (2.1) and hypothesis (H1) since the trace operators W 1,p (Ω) → L r (∂Ω, ρ) and W 1,p (Ω) → L q (∂Ω, ρ) are compact (remember that r, q < p * ). Hypotheses (H2) and (H3) are well known properties of the p-laplacian operator, c.f. [START_REF] Struwe | Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems[END_REF].

Let us consider the compact embedding W 1,p (Ω) → Y = L p (Ω) and denote • p the Lebesgue norm of L p (Ω). The eigenvalue λ 1 defined in (5.3) takes the following expression:

λ 1 := inf Ω (|∇u| p + V (x)|u| p ) dx ; u ∈ W 1,p (Ω), u p = 1
and it corresponds to the least eigenvalue µ of the following eigenvalue problem with Newman boundary conditions :

-∆ p u + V (x)|u| p-2 u = µ|u| p-1 u in Ω, |∇u| p-2 ∂u ∂ν = 0 on ∂Ω. (6.2)
This problem should be understood in the weak sense, that is, for all ϕ ∈ W 1,p (Ω)

Ω (|∇u| p-2 ∇u∇ϕ + V (x)|u| p-2 uϕ) dx = µ Ω |u| p-2 uϕ dx. (6.3)
It is known (c.f. [START_REF] Cuesta | Weighted eigenvalue problems for quasilinear elliptic operators with mixed Robin-Dirichlet boundary conditions[END_REF]) that λ 1 is simple and isolated in the sense that inf{µ > λ 1 ; µ solves (6.3) for some u ∈ W 1,p (Ω) \ {0}} > λ 1 .

We then have that hypothesis (H3)' is satisfied and that E λ1 is of dimension 1. We will denote by ϕ 1 the unique eigenfunction of L p -norm equal to 1 associated to λ 1 . It is also known that ϕ 1 is sign definite and never vanishes in Ω. Furthermore, there is a second eigenvalue λ 2 for problem (6.2) and it can be characterized as

λ 2 = inf{µ > λ 1 ; µ solves (6.
2) for some u ∈ W 1,p (Ω) \ {0}}. (6.4)

In order to apply Theorem 5.7 we are going first to determinate under which conditions on a, b, V the Nehari sign-sets used in theorems 3.3, 3.4, 4.2, 4.1 are non empty. For this purpose we will use Proposition 2.4 of section 2. Let us denote

Γ ± a := {ρ ∈ ∂Ω ; a(ρ) ≷ 0}, Γ ± b := {ρ ∈ ∂Ω ; b(ρ) ≷ 0}, Γ a,0 := {ρ ∈ ∂Ω ; a(ρ) = 0}, Γ b,0 := {ρ ∈ ∂Ω ; b(ρ) = 0}. Lemma 6.1. (1) E + = ∅. (2) If Γ + a = ∅ then N + ∩ E + = ∅. (3) If Γ + b = ∅ then N -∩ E + = ∅. (4) If Γ - a ∩ Γ - b = ∅ then N + ∩ A -= ∅ and N -∩ B -= ∅.
Proof. The proof of ( 1) is trivial and we only prove (2), the proofs of the other cases are similar.

(2) We distinguish two cases: Case

a) Γ + a ∩ (Γ - b ∪ Γ b,0 ) = ∅. In this case we can construct a C ∞ function v in ∂Ω such that A(v) > 0 ≥ B(v). Let u ∈ W 1,p (Ω) having v as its trace. Let ξ be a C ∞ cut function such that 0 ≤ ξ ≤ 1, ξ ≡ 1 in a small ball B(x 0 , r) ⊂ Ω where |u| > c for some c > 0, ξ ≡ 1 in a neighbourhood Ω δ := {x ∈ Ω ; dist (x, ∂Ω) < δ} of ∂Ω, and ξ = 0 in Ω \ B(x, 2r) ∪ Ω 2δ . We can assume that c p B(x0,2r) |∇ξ| p ≥ B(x0,2r) |ξ| p (|∇u| p + V |u| p ) (6.5)
which implies that E(ξu) > 0. Thus ξu ∈ A + ∩B - 0 ∩E + and from Proposition 2.4 (i) we infer that

N + ∩ E + = ∅. Case b) If Γ + a ⊂ Γ +
b the construction of u and ξ runs similarly, starting with v ∈ C ∞ (∂Ω) such that A(v), B(v) > 0. The cut function ξ can be chosen in such a way that (6.5) is satisfied as well as

A(ξu) = A(v) < q -p q -r p -r q -r p-r q-p E(ξu) q-r q-p B(ξu) p-r q-p = max t>0 m ξu (t).
of [START_REF] Lieberman | Boundary regularity for solutions of degenerate elliptic equations[END_REF] gives that they are of class C 1,α (Ω) for some α ∈ (0, 1). Finally the Strong Maximum principle of [START_REF] Vazquez | A strong maximum principle for some quasilinear elliptic equations[END_REF] insures that non negative solutions of the problem are > 0 on Ω.

Remark 6.4. Unfortunately we are not able to replace the conditions j 0 (a) > 0 and j 0 (b) > 0 by, say, a condition related to some suitable eigenvalue. We just remark that both j 0 (a) > 0 and j 0 (b) > 0 imply that λ D 1 > 0, where λ D 1 denotes the first eigenvalue of -∆ p + V |u| p-2 u with Dirichlet boundary conditions. Remark 6.5. In the case a ≥ 0 (resp. b ≥ 0) and a "nice" zero set Γ a,0 one should be able to relate the condition j 0 (a) > 0 (resp j 0 (b) > 0) with the positivity of the first eigenvalue of -∆ p u + V |u| p-2 u over W 1,p (Ω, Γ a,0 ) := {u ∈ W 1,p (Ω) ; u = 0 on ∂Ω \ Γ a,0 }, as was done in [10, Proposition 11].

Existence and multiplicity results for Problem II

Let Ω ⊂ IR N be a bounded smooth domain of class C 2,α (0 < α < 1) with outward unit normal ν on the boundary ∂Ω. Let a, V ∈ L ∞ (Ω) and b ∈ C s (∂Ω) for some s ∈ (0, 1) be possibly indefinite and the exponents r, q to satisfy 1 < r < p < q < p * where p * = p(N -1) (N -p) + and let the number λ be a positive parameter. Take X = W 1,p (Ω) with the usual Sobolev norm • 1,p , the operators

E(u) = Ω (|∇u| p + V (x)|u| p ) dx, A(u) = λ Ω a(x)|u| r dx, B(u) = ∂Ω b(ρ)|u| q dρ,
and the energy functional

J(u) = 1 p E(u) - 1 r A(u) - 1 q B(u) = = 1 p Ω (|∇u| p + V (x)|u| p ) dx - λ r Ω a(x)|u| r dx - 1 q ∂Ω b(ρ)|u| q dρ.
A, B satisfy (2.1) and hypothesis (H1) since the embedding W 1,p (Ω) → L r (Ω) and the trace operator W 1,p (Ω) → L q (∂Ω, ρ) are compact (remember that r, q < p * < p * ). Solutions of Problem II are understood in the weak sense, that is,

Ω |∇u| p-2 ∇u∇ϕ + V (x)|u| p-2 uϕ = Ω λa|u| r-2 uϕ dx + ∂Ω b|u| q-2 uϕ dρ, (6.6
) for all ϕ ∈ W 1,p (Ω). Similar to the previous case, (H2) to (H3)" are also satisfied with λ 1 defined in (6.2) and λ 2 as in (6.4). Let us consider again the compact embedding W 1,p (Ω) → Y = L p (Ω) so the eigenvalues λ 1 and λ 2 have been already defined for Problem I. Let us denote

Ω ± a := {x ∈ Ω ; a(x) ≷ 0}, Γ ± b := {ρ ∈ ∂Ω ; b(ρ) ≷ 0}.
We study now the sign-sets associated to J. Lemma 6.6. (1) E + = ∅.

(

) If Ω + a = ∅ then N + ∩ E + = ∅. (3) If Γ + b = ∅ then N -∩ E + = ∅. (4) If Ω - a = ∅ and Γ - b = ∅ then N + ∩ A -= ∅ and N -∩ B -= ∅ . Proof. (2) 2
We can easily construct a C ∞ ∩ W 1,p 0 (Ω) function ξ with support on a small ball where a > 0 such that E(ξ) > 0.

(3) Let v ∈ C ∞ be a function defined in ∂Ω such that B(v) > 0 and let u ∈ W 1,p (Ω) having v as its trace. Let ξ be a C ∞ cut function such that 0 ≤ ξ ≤ 1, ξ ≡ 1 in a small ball B(x 0 , r) ⊂ Ω where |u| > c for some c > 0, ξ ≡ 1 in a neighbourhood Ω δ := {x ∈ Ω ; dist (x, ∂Ω) < δ} of ∂Ω, and ξ = 0 in Ω \ B(x, 2r) ∪ Ω 2δ . We can assume that which implies that E(ξu) > 0. The cut function ξ can be chosen to satisfy also A(ξu) > -q -p q -r p -r q -r p-r q-p (-E(ξu))

q-r q-p (-B(ξu))

p-r q-p = min t>0 m ξu (t).

Thus ξu ∈ Λ -. The proof of the other case is similar.

We keep here the same notation for the different coerciveness constants, although they read now as follows which implies that E(ξv) > 0. Thus ξu ∈ A + ∩B - 0 ∩E + and from Proposition 2.4 (i) we infer that N + ∩ E + = ∅. b) If Ω + a ⊂ Ω + b the construction of v and ξ is analogous to the previous case, starting with v ∈ C ∞ Ω) such that A(v), B(v) > 0 with support in a small ball B(x 0 , 2r) ⊂ Ω + a . The cut function ξ can be chosen in such a way that (6.9) is satisfied as well as A(ξv) < q -p q -r p -r q -r p-r q-p E(ξv)

q-r q-p B(ξv)

p-r q-p = max t>0 m ξv (t).

The proof of the other cases are similar.

The coerciveness constants are now 1 < 0, Ω bϕ q 1 < 0 then there exist at least two solutions for any λ ∈ (0, µ -1 + ). (ii) Assume that Ω + a = ∅. If c = λ 1 and Ω aϕ r 1 < 0 then there exists at least one solution for any λ ∈ (0, µ -1 + ). (iii) Assume that Ω - a ∩ Ω - b = ∅. If j 0 (a) > 0 and j 0 (b) > 0 then there exist at least two solutions in E -for any λ ∈ (0, µ -1 -). (iv) Assume that Ω + a = ∅. If λ 1 < c < λ 2 , j 0 (a) > 0 and Ω aϕ r 1 < 0 then there exists at least one solution in E + for any λ ∈ (0, µ -1 + ) (v) Assume that Ω + a = ∅ and Ω + b = ∅. If λ 1 < c < λ 2 , j 0 (a) > 0, j 0 (b) > 0, Ω aϕ r 1 < 0, and Ω bϕ q 1 < 0 then there exist at least two solutions in E + for any λ ∈ (0, µ -1 + ). Corollary 6.12. Assume that Ω + a = ∅, Ω + b = ∅ and Ω - a ∩ Ω - b = ∅. If λ 1 < c < λ 2 , j 0 (a) > 0, j 0 (b) > 0, Ω aϕ r 1 < 0, and Ω bϕ q 1 < 0 then there exist at least 4 solutions for all λ ∈ (0, min{µ + ) -1 , µ -1 -}). Remark 6.13. Notice that we do not claim in the statement of Theorem 6.11 that the solutions are positive. The reason is that we can not use, for a local minimizer u ∈ W 2,p (Ω), the relation J(u) = J(|u|) to deduce positivity of solution, since it can happen that |u| ∈ W 2,p (Ω) even if u ∈ W 2,p (Ω). The

  Case I : E(u) > 0 and B(u) > 0 in Figure 1(a); Case II : E(u) ≥ 0 and B(u) ≤ 0 in Figure 1(b); Case III: E(u) ≤ 0 and B(u) ≥ 0 in Figure 1(c); Case IV: E(u) < 0 and B(u) < 0 in Figure 1(d).

Figure 1 .

 1 Figure 1. Possible forms of m u

Figure 2 .

 2 Figure 2. Possible forms of J u

4 .

 4 Local minimizers of J restricted to the Nehari set and to E -

4. 1 .

 1 Minimizing J along N +

4 .

 4 there exist at least two solutions of J (u) = 0 in E + . Proof. (i) Clearly we have (H4), (H5) and (H6) from Proposition 5.5. Thus the local minimum of Theorem 3.3 provides us a first solution of J (u) = 0. Since we also have (H6), a second solution comes from Theorem 3.4. (ii) We have (H4) and (H5) from Proposition 5.5 (a)-(b). Thus the local minimum of Theorem 3.3 is a solution of J (u) = 0. (iii) From (a) and (b) of Proposition 5.6 the hypotheses (H7), (H8) and (H9) hold. Then we get two solutions from Theorem 4.1 and Theorem 4.2. (iv) From (c) of Proposition 5.6 the hypotheses (H4) and (H5) hold. Then we get one solution from Theorem 3.3. (v) From (c) and (d) of Proposition 5.6 the hypotheses (H4), (H5), (H6) and (H7) hold. Then we get two solutions from Theorem 3.3 and Theorem 3.Remark 5.9. As in Proposition 5.5 one has

  ,2r) |∇ξ| p ≥ B(x0,2r) |ξ| p (|∇u| p -V |u| p ) which implies that E(ξu) > 0. Thus ξu ∈ B + ∩ E + and from Proposition 2.4 (i) we infer that N + ∩ E + = ∅. (4) We prove that Λ -= ∅. Let 0 ≤ v ∈ C ∞ be a function defined in ∂Ω such that B(v) < 0 and let u ∈ W 1,p (Ω) having v as its trace. We can assume that u ≥ 0 by replacing u by |u| if necessary. If A(u) ≥ 0 let us take a function 0 ≤ w ∈ W 1,p 0 (Ω) with support in Ω - a such that Ω - a aw r < -Ω au r which implies that A(u + w) < 0. Hence replace u by u + w, which also has v as trace. Let ξ be a C ∞ cut function such that 0 ≤ ξ ≤ 1, ξ ≡ 1 in a small ball B(x 0 , r) ⊂ Ω - a where |u| > c for some c > 0, ξ ≡ 1 in a neighbourhood Ω δ := {x ∈ Ω ; dist (x, ∂Ω) < δ} of ∂Ω, and ξ = 0 in Ω \ B(x, 2r) ∪ Ω 2δ . We can assume that c p B(x0,2r) |∇ξ| p ≥ B(x0,2r) |ξ| p (|∇u| p -V |u| p )

2 Ω

 2 j 0 (a) := inf Ω |∇u| p + V (x)|u| p ; Ω a|u| r = 0, u 1,p = 1 , j 0 (b) := inf Ω |∇u| p + V (x)|u| p ; ∂Ω b|u| q = 0, u 1,p = 1 .B(x 0 , r) ⊂ Ω where |v| > for some > 0, ξ ≡ 0 in Ω \ B(x 0 , 2r). We can assume that p B(x0,2r) |∆ξ| p ≥ B(x0,2r) |ξ| p (|∆v| p + c|v| p ) -|∇v| p |∇ξ| p (6.9)

j 0

 0 (a) := inf Ω |∆u| p -c|u| p ; Ω a|u| r = 0, u W = 1 , j 0 (b) := inf Ω |∆u| p -c|u| p ; Ω b|u| q = 0, u W = 1and one also has to rewrite the constants µ ± in terms of the operators E, A, B. By applying Theorem 5.7 we get Theorem 6.11. (i) Assume that Ω + a = ∅ and Ω + b = ∅. If either c < λ 1 or c = λ 1 and Ω aϕ r

Acknowledgments. This work was partilly carried out while the first author was visiting the IMSP of the Université d'Abomey Calavi (Porto-Novo) and also while the second author was visiting the Université du Littoral Côte d'Opale (ULCO) and the Université Libre de Bruxelles (ULB). We would like to express our gratitude to those institutions.

Proposition 5.5. Let us assume (H1) to (H3) and λ 1 = 0. Then (a) E λ1 ∩ A + 0 ∩ B 0 = ∅ ⇒ (H4), (b) E λ1 ⊂ A -⇒ j 0 (A) > 0, i + (A) > 0 ⇒ j 0 (A) > 0 and (H5), (c) E λ1 ⊂ B -⇒ j 0 (B) > 0, i + (B) > 0 ⇒ (H6).

Proof. We only prove (b), the proof of the other cases are similar. Trivially i + (A) ≥ λ 1 = 0 and j 0 (A) ≥ λ 1 = 0. Assume by contradiction that i + (A) = 0 and let u n ∈ X with A(u n ) = 1 be a minimizing sequence for i + (A). If the sequence u n is bounded then, up to a subsequence, u n u 0 for some u 0 ∈ X. Hence A(u 0 ) = 1 (so in particular u 0 ≡ 0) and E(u 0 ) = i + (A) = 0. Hence

is an eigenfunction associated to λ 1 and we have again a contradiction with the assumption of (b). Thus the sequence (u n ) is unbounded. Let us take v n := v n u n ; hence there exists v 0 ∈ X such that, up to a subsequence, v n v 0 , A(v n ) → 0 and E(v n ) → 0. Notice that v 0 = 0 because otherwise it will follows from 0 = E(v 0 ) ≤ lim inf E(v n ) = 0 and (H3) that v n → 0. This is impossible since v n = 1. We then have v 0 = 0, A(v 0 ) = 0 and E(v 0 ) ≤ 0. By using the inequality

we deduce that E(v 0 ) = 0 so v 0 belongs to E λ1 , a contradiction with the assumption of (b). The proof of j 0 (A) > 0 rules similarly.

We can obtain two more solutions of J (u) = 0 in the case λ 1 < 0, that is, when E -= ∅. To do so, let us assume in this case that

We have

Proposition 5.6. Let us assume hypothesis (H1) to (H3). Then (a) j 0 (A) > 0 ⇒ (H8) and (H9), (b) j 0 (B) > 0 ⇒ (H4) and (H7).

Let us assume also hypothesis (H3) and (H3) . Then (c) j 0 (A) > 0 and E λ1 ⊂ A -⇒ (H4) and i + (A) > 0 ⇒ (H4) and (H5), (d) j 0 (B) > 0 and E λ1 ⊂ B -⇒ (H4) and i + (B) > 0 ⇒ (H4) and (H6).

Proof. (a), (b) are trivial. We only prove (c) as the proof of (d) is similar. Let us denote d := inf{E(u) ; A(u) ≥ 0, u Y = 1} and prove that d > 0. Clearly d > 0 ⇒ (H4). Since for all u ∈ X satisfying A(u) = 1 we have E( u u Y ) ≥ d, the conclusion i + (A) > 0 will follow, and also (H5). First we claim that d is achieved. Indeed, if u n is an admissible sequence with E(u n ) → d then we can prove that the sequence is bounded (otherwise

A variational characterization of λ ± *

We have proved in the previous section the existence of four solutions of the equation J (u) = 0 provided λ ± * > 1, where λ ± * has been defined in (2.8) and (2.9). Let us here give some variational characterization of these values. For the sake of simplicity let us denote α := p -r q -r , β := q -p q -r ,

Observe that

Proposition 5.8. Let us define

If j + 0 (A, B) > 0 then λ + * is achieved. Furthermore, for any u ∈ X where λ + * is achieved we have

A similar result holds for λ - * under the constraint j - 0 (A, B) > 0.

Proof.

is a minimizing sequence for λ + * and (u n ) is bounded then, up to a subsequence, u n u 0 for some u 0 ∈ X that will satisfy A(u 0 ) β B(u 0 ) α = 1, so u 0 will be admissible in the infimum and then it is achieved. If u n goes to +∞ then for v n = un un we will have, up to a subsequence,

un p = 0. We can rule out the possibility

Testing this identity at u we have

that is pE(u) = µ(rβ + qα) = µp and the identity (5.6) follows after a simple computation.

Thus ξu ∈ Λ + and the conclusion follows from (i) of Proposition 2.4.

We remember here the coerciveness values defined in (5.1), that in our case will be j 0 (a) := inf

Finally let us recall the definitions (2.8) and (2.9) and write for sake of simplicity

and

which do not depend on λ.

We can now reformulate Theorem 5.7 as the following existence and multiplicity result for Problem I. Theorem 6.2. (i) Assume that Γ + a = ∅ and Γ + b = ∅. If either λ 1 > 0 or λ 1 = 0 and ∂Ω aϕ r 1 < 0, ∂Ω bϕ q 1 < 0 then there exists at least two solutions for any λ ∈ (0, µ -1 + ). (ii) Assume that Γ + a = ∅. If λ 1 = 0 and ∂Ω aϕ r 1 < 0 then there exists at least one solution for any λ ∈ (0, µ -1 + ). (iii) Assume that Γ - a ∩ Γ - b = ∅. If j 0 (a) > 0 and j 0 (b) > 0 then there exist at least two solutions in E -for any λ ∈ (0, µ -1 -). (iv) Assume that Γ + a = ∅. If λ 1 < 0 < λ 2 , j 0 (a) > 0 and ∂Ω aϕ r 1 < 0 then there exist at least one solution in E + for any λ ∈ (0, µ -1

1 < 0, and ∂Ω bϕ q 1 < 0 then there exist at least two solutions in E + for any λ ∈ (0, µ -1 + ). In particular we have from cases (iii) and (v) that

1 < 0, and ∂Ω bϕ q 1 < 0 then Problem I possesses at least 4 solutions for any λ ∈ (0, min{µ -1 + , µ -1 -}). Proof. Proof of Theorem 6.2. The existence of weak solutions in each of the 4 cases have already be done in Theorem 5.7. Since each solution comes as a local minimizer of J along the sign subsets of the Nehari set and all of these subsets are invariant by taking the absolute value of a function u, we can assume that all these critical points are ≥ 0. Besides the result of [START_REF] Cuesta | Weighted eigenvalue problems for quasilinear elliptic operators with mixed Robin-Dirichlet boundary conditions[END_REF]Theorem A.1] implies that all solutions are bounded and the regularity result We left to the reader the expressions of µ ± in terms of the operators E, A, B. We can now formulate an existence and multiplicity result for Problem II. We generalize some results of [START_REF] Garcia-Azorero | A convex-concave problem with a nonlinear boundary condition[END_REF][START_REF] Sabina De Lis | A concave-convex quasilinear elliptic problem subject to a non linear boundary condition[END_REF] where this problem was studied for V ≡ 0, a ≡ b ≡ 1. Theorem 6.7. (i) Assume that Ω + a = ∅ and Γ + b = ∅. If either λ 1 > 0 or λ 1 = 0 and Ω aϕ r 1 < 0, ∂Ω bϕ q 1 < 0 then there exists at least two solutions for any λ ∈ (0, µ -1 + ). (ii) Assume that Ω + a = ∅. If λ 1 = 0 and Ω aϕ r 1 < 0 then there exists at least one solution for any λ ∈ (0, µ -1 + ). (iii) Assume that Ω - a = ∅ and Γ - b = ∅. If j 0 (a) > 0 and j 0 (b) > 0 then there exist at least two solutions in E -for any λ ∈ (0, µ -1 -). (iv) Assume that Ω + a = ∅. If λ 1 < 0 < λ 2 , j 0 (a) > 0 and Ω aϕ r 1 < 0 then there exist at least one solution in E + for any λ ∈ (0, µ -1

1 < 0, and ∂Ω bϕ q 1 < 0 then there exist at least two solutions in E + for any λ ∈ (0, µ -1 + ). The proof of this theorem comes from Theorem 5.7 and the regularity results quoted in the proof of Theorem 6.2.

1 < 0, and ∂Ω bϕ q 1 < 0 then there exists at least 4 solutions of Problem II for any λ ∈ (0, min{µ + ) -1 , µ -1 -}). Remark 6.9. In [START_REF] Garcia-Azorero | A convex-concave problem with a nonlinear boundary condition[END_REF][START_REF] Sabina De Lis | A concave-convex quasilinear elliptic problem subject to a non linear boundary condition[END_REF] the authors also proved non-existence of solutions for large values of λ. To our knowledge this is an open problem in the non coercive case.

Existence and multiplicity results for Problem III

Let us now discuss the solvability of

The open bounded set Ω ⊂ IR N is assumed here to have a Lipschitz boundary and a, b, ∈ L ∞ (Ω) and c ∈ IR. By a solution of this problem we mean a function u ∈ W 2,p (Ω) ∩ W 1,p 0 (Ω) such that

holds for all ϕ ∈ W 2,p (Ω) ∩ W 1,p 0 (Ω). Here V, a, b ∈ L ∞ (Ω) and 1 < r < p < q < p * * , where

is the critical Sobolev exponent for W 2,p (Ω). The space W (Ω) := W 2,p (Ω) ∩ W 1,p 0 (Ω) endowed with the equivalent norm u W := ∆u p is reflexive uniformly convex Banach space. Take X = W (Ω),

and the energy functional

It is clear that A, B satisfy (2.1) and hypothesis (H1) since the embedding W (Ω) → L r (Ω) and W (Ω) → L q (Ω) are compact (because r, q < p * * ). Thus (H1), (H2) and (H3) hold.

The coerciveness of E depends on the position of c with respect to the first eigenvalue of the following p-bilaplacian problem

with weak formulation

According to [START_REF] Drabek | Global bifurcation result for the p-biharmonic operator[END_REF] there exists λ 1 > 0 a least eigenvalue of (6.8), and this eigenvalue is principal, isolated and simple. Furthermore it holds

Let ϕ 1 > 0 with ϕ 1 p = 1 be an eigenfunction associated to λ 1 . Denote by λ 2 = inf{λ > λ 1 ; λ solves (6.8)}.

The fact that λ 2 ∈ IR is an eigenvalue of the Dirichlet p-bilaplacian and the existence of a sequence of eigenvalues is proved for instance in [START_REF] Talbi | On the Spectrum of the Weighted p-Biharmonic Operator with Weight[END_REF].


Christophe Benech
email: christophe.benech@mom.fr

New Approach to the Study of City Planning and Domestic Dwellings in the Ancient Near East

Keywords: magnetic survey, city planning, domestic unit, space syntax, Doura-Europos

This paper presents the results of a magnetic survey on the Hellenistic and Roman site of Doura-Europos in Syria. The interpretation of the magnetic data is based on an original approach by considering the use of space in a domestic unit. This type of study has been developed for sociological research but is adapted to the information carried within geophysical data. After a brief presentation of the role of geophysical methods for the study of city planning, the most important components of the 'space syntax' will be presented and applied to two blocks of Doura-Europos, one that has been excavated in the twentieth century by the Yale University and another surveyed using the magnetic method.

INTRODUCTION

The different geophysical methods available today allow the collection of very detailed information about archaeological sites and their environment. However, these spectacular results are still often confronted with scepticism and even with disbelief by archaeologists (Hesse, 1999;[START_REF] Gaffney | Project specifications or guidelines in commercial archaeological geophysics: a recipe for disaster? 6 th International Conference on Archaeological Prospection[END_REF] because of the problem of interpretation and of integration of geophysical data into archaeological maps. This is a subject that has been discussed many times and interesting solutions have been proposed (see [START_REF] Picard | GIS in Archaeology -the interface between Prospection and Excavation[END_REF].

A geophysical map is not an archaeological plan and it would be utopian to expect the precision of a topographical plan of an excavated areathe base map from which archaeologists workfrom it. These two sources provide very different types of information, which cannot be treated in the same way. If we treat a geophysical map in the same way as an archaeological plan, the result will always be less reliable, and the essential and original contribution of the geophysical data will be neglected. It is more fruitful to focus on their complementary nature in order to provide a new vision of the archaeological site and renew thematic and methodological approaches.

For better utilization of the geophysical map in archaeology, geophysicists and archaeologists have to hold a dialogue concerning (i) the type of information that can enhance knowledge of the archaeological site and (ii) the development of a new approach to archaeological issues arising from documentation based on architecture, stratigraphy and, more generally, the spatial organization of archaeological remains. This is the sort of process is developed here, in the particular case of city planning and households in the ancient Near East. The present study is based on geophysical surveys carried out at the site of Doura-Europos (Syria), where extensive documentation concerning urbanism and households exists. In this case, data from a magnetic survey are used, but the more general discussion could extend to other geophysical methods.

THE STUDY OF HELLENISTIC AND ROMAN URBANISM IN ARCHAEOLOGY

The study of city planning, and more generally urbanism, is best undertaken by an extensive archaeological survey, the aim of which is to recognize the urban space as completely as possible. Hellenistic cities of the Near East adopted the same the city planning, which is known as the 'Hippodamian plan'. This concept, brought by Macedonian settlers, is based on the parcelling of rectangular blocks separated by orthogonal streets of the same width (Figure 1). These blocks were also divided into dwelling units of the same size. The dimensions of the blocks, their inner division and the width of the streets varied from one site to another but theoretically are supposed to be the same for a given city [START_REF] Picard | GIS in Archaeology -the interface between Prospection and Excavation[END_REF]. The Hippodamian plan is therefore taken to illustrate a particularly egalitarian -restrained and austere -division of the urban space, as opposed to the more monumental and hierarchical urbanism developed by the Romans.

Until recently, the method used to study the urban space was almost the same for all the archaeological sites of the Near East, and it has also been applied in Doura-Europos (Figure 2). Wide-area excavation at the beginning of the twentieth century at some ancient cities resulted in a rich archaeological documentation. Some blocks were completely excavated, and the entire town plan was reconstructed by digging up the angles of non-excavated blocks (when it was possible, as in Doura-Europos) or by extrapolation, reproducing on the whole surface of the city the module of reference identified in excavated blocks. Those early excavations were focused on 'prestigious' monuments: administrative or religious buildings, rather than domestic dwellings. Apart from some sites such as Olynthus (Greece), Delos (Greece) and Doura-Europos, domestic dwellings were usually illustrated by the excavation of a 'standard block' which was supposed to represent all the households of the city.

Therefore, this method only allowed for the reconstitution of a schematic plan of the city. It may be possible in this way to illustrate the theoretical concept used for the organization of the city, but it cannot help us understand the real administration of the urban space -and its evolution. For reasons of cost, and also because of the emergence of a better methodology of research and the question of conservation of excavated remains, such large-scale excavations are no longer undertaken. Consequently, current research, suffers because of a drastic lack of new data. Modern excavations, more punctual but also more rigorous, yield important data about the life of the city, but do not permit us to refine the simplified scheme of the organization of the urban space. Geophysical survey is currently the only non-destructive technique for obtaining new data. The results can be more interesting than the data obtained hitherto, because if the environmental conditions are good enough the whole urban layout of a city can be investigated. Numerous such studies have been made or are still in progress (e.g. [START_REF] Becker | Magnetische Prospektion in der Untersiedlung von Troia[END_REF][START_REF] Gaborit | Zeugmamoyenne vallée de l'Euphrate : rapport préliminaire de la campagne de fouille 1998[END_REF][START_REF] Groh | Integrated prospection in the upper town of Ephesus, Turkey -a case study, 5 th International Conference on Archaeological Prospection[END_REF][START_REF] Schmidt-Colinet | Geophysical survey and excavation in the "Hellenistic Town" of Palmyra[END_REF]. These surveys have helped to highlight the main characteristics of the urban space of different ancient cities, i.e.:

(i) limits of the city, fortifications, location of gateways;

(ii) main characteristics of the plan -length and orientation of the streets, block module;

(iii) differentiation between built and non-built areas;

(iv) differentiation between public and private areas;

(v) identification of the function of some buildings that have a typical plan.

These results are important in themselves but do not account for the richness of geophysical data, because only the most visual and the most readable information from the geophysical map is used. Mostly, it is at this step of the interpretation that dissent occurs between archaeologists and geophysicists, because the geophysical map lacks the precision of an archaeological plan despite its detailed information. The detailed interpretation of the geophysical map is normally limited to a 'line by line' restoration of visible geophysical anomalies; the result should be a noninterpretative drawing but this type of work reduces the coherence of built structures-it involves conscious choices and, moreover, unconscious choices.

Different experiences of the interpretation of geophysical data have shown that, whatever the type of reconstruction chosen, it can be applied only when the logic and the organization of the archaeological features being analyzed are clearly identified, and understood, by a more detailed interpretation (e.g., in the case of city planning and buildings, see [START_REF] Hesse | Introduction géophysique et notes techniques[END_REF][START_REF] Groh | Geophysikalische Messungen im NordÖstlichen Stadtteil von Flavia Solva: Interpretation und archäologische-historische Auswertung[END_REF].

A detailed interpretation must first of all be based on archaeological documentation of the site (if available) or from another site from the same period and with the same type of occupation. From this point of view, it is obvious that archaeological and geophysical approaches are closely linked: the work of the geophysicist does not end with the presentation of more or less detailed maps, and that of the archaeologist does not begin with the archaeological interpretation of this presentation. A new step may be suggested, departing from our conviction that every interpretative approach is linked to the problems posed by archaeology. By contrast, it is sometimes necessary to reorient the questions of the archaeologist, in order to analyse the geophysical data with the most appropriate method.

This type of approach is applied here to the households of the Near East Hellenistic city of Doura-Europos.

DOURA-EUROPOS (SYRIA)

The site of Doura-Europos was discovered at the beginning of the twentieth century and was excavated by F. Cumont (1922-23), a French and American mission directed by M. Rostovtzeff (Yale University), and since 1986 a French and Syrian mission directed by P. Leriche (Centre National de la Recherche Scientifique). Doura-Europos is located on the right bank of the River Euphrates on a limestone plateau at around 30 m above the river (Figures 2 and3). The historical developments outlined below highlight the most important phases of the urban development of the site as they have been understood by successive archaeological excavations.

Founded at the end of the fourth century BC by the Macedonians, Doura-Europos was originally a military station. The concept of the city planning (and therefore the accession to the status of 'city'), began in the middle of the second century BC. About 133 BC, the city fell under Parthian domination; by this date, only the sector of the agora and also some administrative and religious buildings had been completed. The new Parthian authorities continued the urban programme respecting the Hippodamian plan that had been established by the Macedonians. The dimensions of the blocks are on average 35 m x 70 m; the width of the streets is about 5 m, except for the main streets, starting from the Palmyra gate, which reach about 10 m. In AD 165, the city was taken over by the Romans, and the city planning was subsequently greatly modified, mainly due to the installation of a military camp in the northern part of the city. The Romans constructed many buildings in this sector without taking the pre-existing Hippodamian plan into account. Finally, in AD 256, Doura-Europos was pillaged by the Sassanids and practically abandoned. Obviously, the geophysical map shows the last phase of more than four centuries of occupation i.e. the result of the evolution of the application of an urban theoretical concept: the Hippodamian plan.

FIRST RESULTS WITH MAGNETIC SURVEY

The magnetic survey of Doura-Europos was carried out with a caesium gradiometer, and the southern half of the site has been completely covered (Figures 2 and4) [START_REF] Benech | The study of ancient city planning by geophysical methods: the case of Doura-Europos, Syria. 5 th International Conference on Archaeological Prospection[END_REF].

Unlike the schematic plan established by the Yale expedition (1928)(1929)(1930)(1931)(1932)(1933)(1934)(1935)(1936)(1937), the geophysical data record variations street width. In some cases the variation is considerable and indicates a hierarchical organization of the streets, probably linked to the circulation inside the city. Some of the main streets (particularly the ones starting from the Palmyra gate) conserved their original width, whereas others (probably those less frequently used) became narrower with time due to the encroachment of houses onto the public thoroughfare.

According to the results of the excavations, the Yale expedition suggested an internal division of the blocks into eight identical parcels, although this division is not really clear in the excavated blocks. The geophysical map clearly identifies some plans of houses that were already known from the excavations, and which fill exactly 1/8 of the total area of the block (Figure 5). Not all blocks of Doura-Europos, however, have the same internal division: in the southern part of the city the geophysical map reveals two blocks (I 10 and I 11) that are divided into six parcels. This is the first time that the existence of two different types of division has been observed in a Hellenistic city of the Near East, but we do not know yet if they are contemporary. In block I 11, two of these parcels are free from construction, and different hypotheses may be advanced concerning their function: they could be gardens, stock areas or enclosures for animals, a notion which might be justified by the proximity of the southern gate of the city.

THE STUDY OF DOMESTIC SPACE

The excavations of Doura-Europos present a pronounced diversity of the ground plans of domestic houses. The geophysical maps, even if they are very clear, do not identify the plan of every house inside the blocks. There are always numerous ambiguities, in particular concerning the limits of the houses, the function of the different rooms and the circulation inside the house (to which may be added the problems of blocked doors (Figure 6) or partly ruined walls, which give the illusion of a passage on a geophysical map). These aspects are, of course, all easy to solve in excavation and play an important role in the study of the household following the traditional approach.

The study of individual households traditionally has been based on the classical literature available. However, written sources are of little value with respect to the archaeological data, which are mostly fragmentary. They describe a way of life very localized geographically, and leave the door open to misleading generalizations (for a critique of the use of classical sources see the introduction of [START_REF] Nevett | House and Society in the Ancient Greek World[END_REF].

Study of the household can be approached from Figure 5. Comparison between the plan of an excavated house (from Hoepfner and Schwander,1986) and the plan of a house shown by geophysical survey (from block M4 -minimum white=-10nT/m: maximum black=+10nT/m).The entrance is not on the same side of the block but there is a corridor, which means that the courtyard is not visible from the street. In the southern part of the house, there is the reception room, opened to the north, with two adjacent rooms. The eastern side of the house is covered but open to the courtyard and is the place of domestic activities such as cooking. This is a typical domestic unit of Doura-Europos, which occupies exactly 1/8 of the surface of the block. (E, entrance; C, courtyard; RR, reception room).

another point of view, which is the use of space. This method has been used previously by archaeologists, in particular those working on the most ancient periods [START_REF] Hodder | Spatial analysis in archaeology[END_REF][START_REF] Renfrew | Approaches to social archaeology[END_REF]. Their aim was to introduce methods of analysis developed in ethnology, sociology and anthropology [START_REF] Rapoport | Pour une anthropologie de la maison[END_REF][START_REF] Levi-Strauss | Tristes tropiques[END_REF][START_REF] Bourdieu | La maison ou le monde renversé. In Esquisse d'une théorie de la pratique[END_REF]. Important works have shown that the study of the use of domestic space is complementary to a more traditional architectural study (Kent, 1990a;Parker Pearson and Richards, 1994;Allison, 1999). Some of these studies are concerned with the Classical period [START_REF] Jameson | Domestic space in the Greek City State[END_REF]Laurence and Wallace-Hadrill, 1997;Nevet 1999;[START_REF] Cahill | Household and city organization at Olynthus[END_REF] and they have yielded interesting, albeit at times somewhat fragmentary, results in accordance with the archaeological documentation available for the different sites that have been studied. Sociologists B. Hillier and J. Hanson have shown that the organization of space may reflect the social, economic and cultural characteristics of a society. They developed the 'space syntax' method of analysis, which can be easily applied to geophysical maps (Hillier and Hanson, 1984; for a summary see [START_REF] Grahame | Public and private in the Roman house: the spatial order of the Casa del Fauno[END_REF]. The geophysical map brings a new, continuous and homogeneous documentation to this domain, which can play an important role if the methodology is adapted to the nature of geophysical data. The archaeological studies cited above present the idea that it is the use of space that influences the record of archaeological materials (ceramics, objects, etc.) and their architectural elements -not the other way around (Kent, 1990b). Such an approach is interesting to apply to geophysical maps, even if it is necessarily limited to a study of the ground plan, which admittedly could be considered to be extremely restricted. The chosen approach is to use as a starting point a plan delimiting all the spaces, and not from a plan locating the walls. Such a reconstruction is more accurate given the nature of geophysical data: it is very difficult indeed to restore the geometry of the walls with precision, to the same degree as that inherent in a plan derived from excavation. On the other hand, it is not hard to delimit surfaces. Moreover, the quality of a study of the use of space does not suffer from a certain error tolerance that is inevitable if we work from a geophysical map.

For the first time, this approach will be validated by a comparative study of two blocks of Doura-Europos, one completely excavated by the Yale mission, and another one surveyed with the caesium G-858 gradiometer. To fulfil this aim the main characteristics of a household are presented in order to understand the logic of the use of space in the houses of Doura-Europos.

MAIN CHARACTERISTICS OF DOURA-EUROPOS HOUSES

The houses of Doura-Europos are organized around three main elements, the entrance (E), the courtyard (C) and what for present purposes will be referred to as 'the reception room' (RR) (Figure 5).

The entrance is normally an 'L' shaped corridor leading to the courtyard, which generally is not visible from the street; but in the case of particularly small houses the entrance looks directly on to the courtyard. There may be more than one entrance, particularly if the house is large; there are also houses where a room is used as a shop, which has an independent entrance. The open courtyard is, generally speaking, the main place of the house. It is the point from which circulation inside the house is organized, and it controls the access to most of the other rooms, and to the roof or the first floor, when one exists. It is the focal point of the main domestic activities (cooking).

The reception room is the most significant room from a social point of view. This room stands out from the rest by its comfort, architectural decoration and orientation: it is generally open to the north, to preserve a minimum of freshness inside the room. However, in the case of smaller houses, the position of the reception room may be different.

The space syntax suggested by Hillier and Hanson (1984) allows visualization of the organization of space inside the houses. Plans that look different may in fact have the same logic of organization. We have modified the scheme proposed by Hillier and Hanson: the circles on Figure 7, which represent individual rooms, are proportional to their surface areas, and the whole scheme is enclosed in a circle proportional to the total surface area of the house.

Figure 7. Scheme of houses 2 and 4 in block C7 (see Figure 9) following the space syntax of Hillier and Hanson (1984). We see here the articulation entrance (green), courtyard (red), reception room (blue), for this last one. The widest houses are first characterized by the greater number of rooms around the courtyard.

When using this scheme, account must be taken of ambiguities due to the interpretation of the magnetic maps:

(i) The circulation inside the house presents a problem: the passages between the rooms are not always clear. Partly ruined walls can be mistaken for an opening. In the case where archaeological structures are preserved on a very weak elevation, the magnetic signal of the stone block (gypsum most of the time), which constitutes a threshold, may be mistaken for the signal of a wall; in such cases, the opening is then practically undetectable.

(ii) The excavated blocks show a pronounced variation in the width of the walls (from 20 cm to more than 1 m); the thinnest dividing walls are usually completely ruined and may not appear on the magnetic map.

(iii) Another difficulty concerns the delimitation of the different houses of the block. This question is all the more complex because transformations through time inside the house are frequently seen in Doura-Europos, mainly due to alterations caused by inheritance [START_REF] Saliou | Les quatre fils de Polémocratès (P. Dura 19)[END_REF] or by the purchase of rooms by neighbouring houses. Houses 4 and 6 of block C7 are highly complex examples of this, and clearly show the type of ambiguity that our study confronts (Figure 8). These houses are examplars of the division of a 'standard' size house (i.e. 1/8 of the surface of the block), which had a central courtyard and distribution of rooms around it. The identification of two different courtyards and two entrances would be obvious to an excavator, but not so on a magnetic map. Such an organization presumably would be interpreted as a unique house.

Figure 8. Plan of houses 4 and 6 in the bock C7 (see Figure 9). The total surface of both houses is equivalent to 1/8 of the surface of the blocks, indicating that an original division has been divided into two dwelling units. Such an organization could appear as a single domestic unit on a geophysical map: only the two entrances suggests that there are two units, but even in this case the second entrance could interpreted as one associated with a shop.

These limitations necessitate an initial argument about the concept of space rather than about the use of space, but a statistical study may help to further the interpretation. Of course, it is impossible to present a reliable statistical analysis on the basis of a study of two blocks only, but the relevance of this application to an understanding of excavated and surveyed blocks clearly emerges, which enables a predictive model to be built that will facilitate the interpretation of geophysical maps and offer an original and new approach to the study of the household.

In archaeological publications, the terms 'house' and 'dwelling unit' are often synonymous. The term 'house' is used in the following interpretation when domestic units have been identified with certainty or with reference to the social functions of a private building; the term 'dwelling unit' will be used for units which potentially include rooms with nondomestic functions (shop, workshop, etc.).

CASE STUDY I: BLOCK C7

Block C7 is the only completely excavated block of Doura-Europos that comprises only domestic dwellings, i.e. no administrative or religious buildings (Figure 9) [START_REF] Hopkins | The houses[END_REF][START_REF] Saliou | La forme d'un îlot de Doura-Europos… L'îlot C7 revisité[END_REF]. There are two other blocks of domestic dwellings, but they constitute particular cases because they comprise particularly large houses, which nearly (block D5) or entirely (D1) occupy the area of the block.

In its final state of occupation, block C7 contained 12 dwelling units, the limits of which no longer correspond to the original division. The entrance, the courtyard and the reception room are clearly identified for all units except for unit 9, where the location of the reception room is uncertain. The area of the dwelling units is extremely variable, from 54 m 2 to 320 m 2 , which is far from the egalitarian division of the Hippodamian concept. Many uncertainties remain about the organization of the block, which was quickly excavated using the techniques current at the beginning of the twentieth century: neither its stratigraphy nor its chronology were recorded well. Some of the rooms continued in use in later periods, and it is difficult to establish their correct plan, even for the last phase of occupation (the plan of unit 9 is particularly doubtful).

There are 17 entrances for 12 dwelling units, hence the block may contain five shops. It is beyond the scope of this paper to apply the space syntax approach to each dwelling unit. The following are the important results for the interpretation of the geophysical map.

(i) The 'entrance-courtyard-reception room' layout is respected in all cases apart from units 7 and 9, where the entrance leads to a room instead of a corridor ii) Generally there were four levels of depth, (number of steps that must be taken to arrive to the deepest room of the house starting from the entrance), whatever the size of the dwelling unit; and at times even a fifth level.

(iii) The main difference is the number of rooms around the courtyard, which increases with the size of the dwelling unit (see e.g. units 2 and 4, Figure 7).

(iv) In most cases the rooms are distributed on three sides of the courtyard. In unit 12, a reception room was lacking. Of a total of 11 reception rooms, six were located in the southern part of the dwelling unit and open to the north.

Figure 10 compares the surface area of individual rooms against the surface area of the dwelling. Except in unit 9, where the identification of the reception room is uncertain, the courtyard and the reception room are the largest rooms of the houses. The reception room is generally larger than the courtyard, and the more important the house, the greater the increase in surface area of these two rooms. On the other hand, the mean surface area of the other rooms is almost the same for all dwelling units. Figure 11 demonstrates the importance of the courtyard and the reception room to domestic life: even in the smallest dwelling unit, number 3, more than 80% of the total surface area is used for these two rooms.

This study is admittedly limited to only one block, but these first results highlight certain interesting social and cultural aspects of the domestic dwelling. It can be assumed that the size of the dwelling unit was proportional to the richness of the occupier, as proved by excavation. Most dwelling units have a surface area of about 200 m 2 , i.e. less than the original 1/8 division (306.25 m 2 ). Unit 9 is the only one to have a surface area >300 m 2 . The courtyard and the reception room were the most important places within the houses. These spaces were the parts of the house a visitor was likely to see, and it was therefore important to enhance them with exterior signs of richness and prestige. Another surprising result is that the other rooms have almost the same surface area, whatever the size of the dwelling unit. Only the number of rooms increased with the size of the dwelling unit. This tendency seems to show that whatever the social class (probably linked to a certain form of richness allowing an individual to have a bigger house), the inhabitants did not feel the necessity to live in larger rooms: private spaces had more or less the same dimensions.

Block C7 demonstrates an important diversity in the richness and size of its dwelling units, but this may be relative, as all of the proprietors may have belonged to the same cultural class and had the same social origin. These are only preliminary results of an approach applied for the first time to the households in Doura-Europos and therefore must be treated with with prudence until they are confirmed by excavation and survey.

CASE STUDY II: BLOCK M2

The general rectangular shape of block M2 and its dimensions hardly changed with time and remain close to the reference model (Figure 12); -although its subdivision into eight units is no longer visible for most of the last stage of its occupation. The lengthwise division can still be observed for the whole length of the block even if the wall does not look continuous. The transverse middle division continues to be visible but no longer seems to function as a separation between dwelling units any more (see below). The field survey map produced by the Yale expedition indicates 11 entrances for block M2: three were located on the western side of the block, one on its southern side and seven on its eastern side. All of these entrances are visible on the magnetic map. The widest of the streets surrounding the block is located along its eastern side, which also has the most entrances, and we can make the same observation as in the case of block C7: an increase in the number of entrances on one side of a block (mainly due to the presence of shops) is linked to the width of the streets. The next step is to delimit the spaces inside the block: as previously stated, the most important aspect is to identify the different spaces, and uncertainty about the geometry or the exact surface area of the space does not affect the study (Figure 12c). The block is accordingly divided into 83 space units. Here, as with block C7, the courtyard and the reception room were the widest rooms of the house. Figure 12d categorizes rooms with a surface area greater than 25 m²: the logical position of the courtyard and the reception room are thereby easily identified. It is now possible to hypothesize the delimitation of the different dwelling units based on the observations of the space syntax schemes of block C7. Thus nine dwelling units can be delimited: four on the eastern and five on the western side. In the southern part of the block, plans of houses 3, 4, 5 and 6 are clearly visible with the classic organization observed in Doura-Europos; units 3 and 4 therefore have two entrances. The northern part of the block is more problematic because it has been subject to important transformations in the organization of space. Nonetheless, the identification of the courtyards and reception rooms allows units 7, 8 and 9 to be delimited. In this hypothesis, only the entrance of unit 8 is not visible on the magnetic map and is not mentioned on the Yale map; the corridor between the supposed entrance and the courtyard is nevertheless clearly visible.

Units 1 and 2 are more complex because of the number of their three entrances and their location. The entrance, the courtyard and the reception room of unit 1 are clearly identified, but its boundary with unit 2 is rather unclear; it has to be located on the line of the southern wall of the reception room, because the northern entrance of unit 2 is immediately to the south. This dwelling unit therefore has two entrances, and it is in fact the widest of the block with four rooms with a surface area greater than 30 m 2 . Either a large dwelling unit existed here -including an independent entrance for a shop -or this area represents two houses. In the latter case the southern house would not have an entrance corridor but a direct access to a room of more than 25 m 2 . DISCUSSION Even though it is not possible to draw definitive conclusions about the use of domestic space in Doura-Europos based on the study of only two blocks, it is interesting to compare the results obtained in both cases.

It was concluded that block M2 is made up of nine dwelling units compared with 12 dwelling units in block C7. The dwelling units accordingly were widest in M2, with the minimum surface area for dwelling units of M2 being >200 m 2 .In contrast there are six dwelling units in C7 with a surface of <200 m 2 . A histogram of the surface area of the rooms (Figure 13) shows that there were more smallsize rooms in C7, but the global distribution of the surface is the same for both blocks.

A graph of the distribution of the surface area of the rooms for each dwelling unit in M2 produces a different result (Figure 10c).

In block M2 the mean surface area of the courtyards is greater than that of the reception rooms. The linear regression line decreases in the case of the courtyard, which is attributable to the size of unit 2. This leads to two possibilities:

(i) there are two houses, as hinted at above;

(ii) unit 2 partly or entirely comprises an administrative or, more probably, a religious building. The latter is commonly seen in Doura-Europos. Many houses were transformed into religious buildings, whereby they maintain some characteristics of the domestic space, but their widest rooms have been transformed to receive the public. If unit 2 is divided into 2a and 2b (Figure 10d), a closer correlation between M2 and C7 is evident, even if the coefficient of determination for M2 (r²=0.22) is very inferior to that of C7 (r² =0.44, or 0.83 if unit 9, which is a doubtful case, is not taken into account). As for the courtyard, the correlation between the reception room and the size of the dwelling unit is less evident in the case of M2 (r²=0.16) than for C7 (r²=0.35, or 0.59 without unit 9).

The correlation between the courtyards, reception rooms and the dwelling unit size is less clear in the case of M2 even if the proportion of surface dedicated to both these rooms is almost the same (Figure 11b). Nevertheless it must be noted that the dwelling units of M2 are on average larger than those of C7, and the correlation observed in C7 is not the same for very large dwelling units, such as unit 2 in M2 and unit 9 in C7.

CONCLUSIONS

The results presented here show that an interpretative 'space syntax' approach adapted to the specific of geophysical data can open rich new avenues of research that are inaccessible by the common methods used in archaeology. The results presented here are, of course, partial, but they show how useful the study of space may be for the interpretation of geophysical maps, and at the same time the potential for developing such an approach in archaeology. A complete study would involve taking into account all excavated and surveyed dwelling units of Doura-Europos and the layout of streets in order to obtain a concerted vision of the organization of the public and private space of the city. From this point on, it will be possible to study the social and cultural aspects in relation to the use of space, at the level of not only the domestic unit but also of the city. Such an approach will considerably modify the study of city planning and will demonstrate that the role of the geophysical map is not only to offer a 'global view' for locating new places to excavate.

Figure 1 .

 1 Figure 1. Hippodamian plan of Miletus (Turkey) known as one of the oldest city plans of this type. It comprises two different modules of blocks: 51.60m x 29.50m for the southern part, and 20.75m x 17.70m for the northeastern part.

Figure 2 .

 2 Figure 2. Plan of Doura-Europos (Syria) (from Rostovtzeff, 1939 and redrawn by H. David). The area surveyed is in dark grey and the excavated areas are shown in middle grey.

Figure 3 .

 3 Figure 3. Aerial view fromthe south-east of the site of Doura-Europos.

Figure 4 .

 4 Figure 4. General magnetic map of the geophysical survey of the southern part of the site (minimum white=-10 nT m -1 : maximum black=+10 nT m -1 ). The magnetic survey has been carried out with a caesium gradiometer recording measurements every 0.20m along profiles of 1m apart.

Figure 6 .

 6 Figure 6. Example of a door blocked with mudbrick.

Figure 9 .

 9 Figure 9. Plan of block C7 (from Saliou, 2005).The block comprises 11 dwelling units but there are 17 entrances, indicating the presence of shops with independent entrances.

Figure 10 .

 10 Figure 10. Distribution of the surface of the rooms for each domestic unit: (a) block C7; (b) block C7 without unit 9; (c) block M2; (d) block M2 with unit 2 divided into domestic units.

Figure 10

 10 Figure 10. (Continued).

Figure 11 .

 11 Figure 11. Ratio of the surface room/house: (a) block C7; (b) blockM2.

Figure 12 .

 12 Figure 12. (a) Magnetic map of block M2. (b) Theoretical division in equal parcels of a block with dimensions 35m_70m. (c) Hypothesized division of the space inside the block. (d) Hypothesized distribution of the courtyards and reception rooms.

Figure 13 .

 13 Figure 13. Histogram of the surface area of the rooms in blocks C7 and M2.