Scott N Armstrong
email: armstrong@math.uchicago.edu

Boyan Sirakov
email: sirakov@ehess.fr

A NEW APPROACH TO LIOUVILLE THEOREMS FOR ELLIPTIC INEQUALITIES

Keywords: Mathematics Subject Classification. Primary 35B53, 35J60, 35J92, 35J47 Liouville theorem, semilinear equation, p-Laplace equation, Lane-Emden system

Introduction

In this article we review a new method for proving the nonexistence of positive solutions of elliptic inequalities in unbounded domains in R N , N ≥ 2, which was recently introduced by the authors [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF]. For clarity and ease of exposition here we consider only domains which contain the infinite point of R N , that is, domains in the form R N \G, where G is an arbitrary bounded set. To further simplify the presentation we are going to state our results for the following simple but important and widely studied inequalities [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF] -∆ p u ≥ f (u) in R N \G, and

P + λ,Λ (D 2 u) ≥ f (u) in R N \G (2)
, where f : (0, ∞) → (0, ∞) is some continuous map. Our results, which are new even for the standard semilinear inequality -∆u ≥ f (u), provide sharp hypotheses on f under which [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF] and [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF] have no positive solutions.

We have taken [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF] as an example of an inequality in divergence form, whose weak solutions are naturally defined in Sobolev spaces. Here -∆ p denotes the p-Laplacian and p > 1.

In (2) P + λ,Λ denotes the Pucci maximal operator P + λ,Λ (M ) = sup λI≤A≤ΛI tr(-AM ), for some positive constants λ ≤ Λ. This equation is in non-divergence form, and its weak solutions are naturally defined in the viscosity sense. Note the non-existence of solutions of (2) implies any semi-linear inequality -a ij (x)∂ ij u ≥ f (u) has no solutions either, provided the eigenvalues of the matrix (a ij (x)) lie in the interval [λ, Λ].

We first state the result we obtain on [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF]. The type of condition we impose on f varies depending on how p compares to N . Theorem 1 ( [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF]). Denote

p * := N (p -1) N -p if p = N. Assume that f : (0, ∞) → (0, ∞) is continuous and satisfies (i) if p < N , then lim inf t→0 t -p * f (t) > 0; (ii) if p = N , then lim inf t→∞ e at f (t) > 0 for each a > 0; and (iii) if p > N , then lim inf t→∞ t |p * | f (t) > 0.
Then the inequality (1) has no positive weak solution.

This theorem is sharp: for instance the model inequalities -∆ p u ≥ u p * +ε have positive solutions in every exterior domain if p < N and ε > 0, or if p > N and ε < 0 (resp.

-∆ p u ≥ e -au has solutions if p = N , for each a > 0; see the end of the paper).

The study of the nonexistence of positive supersolutions of elliptic equations and systems has a rich literature. While we do not give extensive references here, referring instead to the more complete bibliography in our paper [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF] as well as to [START_REF] Véron | Singularities of solutions of second order quasilinear equations[END_REF][START_REF] Mitidieri | A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities[END_REF][START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF][START_REF] Kondratiev | Positive solutions to semi-linear and quasilinear elliptic equations on unbounded domains[END_REF][START_REF] Lorenzo | A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities[END_REF], we do mention that special cases of Theorem 1 have been proved among other things by Gidas [START_REF] Gidas | Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations[END_REF], Ni and Serrin [START_REF] Ni | Nonexistence theorems for quasilinear partial differential equations[END_REF], Bidaut-Veron [START_REF] Bidaut-Véron | Local and global behavior of solutions of quasilinear equations of Emden-Fowler type[END_REF], Bidaut-Veron and Pohozaev [START_REF] Bidaut-Véron | Nonexistence results and estimates for some nonlinear elliptic problems[END_REF], Serrin and Zou [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF] and more recently by d'Ambrosio and Mitidieri [START_REF] Lorenzo | A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities[END_REF]. The previous methods for proving Liouvilletype results like Theorem 1 have involved either assembling delicate integral identities using the integral formulation of the equation or, should the symmetries of the equation permit, "radializing" the equation, that is, showing that the spherical mean of an eventual solution satisfies an ODE without solutions. Our technique, which will be developed below, is rather different and relies on some simple ideas related to the maximum principle.

What is striking about Theorem 1 at first glance is how little is required of the function f . Only local conditions are imposed on the behavior of f , in the sense that we demand only that f (t) either grow fast enough near t = 0 or decay slowly enough near t = ∞, but allow arbitrary behavior elsewhere. In constrast, most of the previous papers considered the case f (t) = t q , q > 0. To our knowledge, only the hypothesis (i) in Theorem 1 has appeared before, for the first time in [START_REF] Ni | Nonexistence theorems for quasilinear partial differential equations[END_REF] for decaying solutions, and recently in [START_REF] Lorenzo | A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities[END_REF] for differential inequalities holding in the whole space R N . The possibility of allowing nonlinearities which decay at infinity in the case p ≥ N has not been observed (except for the trivial case of an inequality in R N where p-superharmonic functions do not exist).

The conditions (i) -(iii) can be best explained if we remember the dilative scaling of the equation in the model case f (t) = t q for q = p -1. As it is easy to check, if u is a solution of (3)

-∆ p u ≥ u q in R N \ B 1 ,
then for any s > 0 the rescaled function u s (x) := s q * u(sx) is a supersolution of the same equation in the domain R N \B 1/s , provided we set the scaling exponent to be q * := p/(q -p + 1).

The question of existence or nonexistence of positive supersolutions of (3) turns out to depend on the competition between this scaling exponent q * and the homogeneity

α * = (N -p)/(p -1)
of the fundamental solution Φ = Φ p (x) of the p-Laplace equation, which we recall is given by

Φ p (x) = ±|x| -α * if α * = 0, Φ p (x) = ± log |x| if α * = 0.
For example, for (3) if q > p -1 and p < N condition (i) is equivalent to the inequality 0 < α * ≤ q * . Similarly if q < p -1 and p > N condition (iii) requires that q * ≤ α * < 0. This point of view also explains why the conditions in Theorem 1 are sharp: to find a supersolution (e.g., in the model case f (t) = t q ) one needs only to slightly modify the fundamental solution Φ p by bending it in an appropriate way. A first discussion on the interplay between α * and q * appeared in our earlier paper [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF], where we used an argument based on a "linearization" to study some particular inequalities in non-divergence form.

Let us now state the result we obtain on the inequality (2). Dividing the inequality by Λ we can assume Λ = 1. We will also assume we are in the non-trivial case λ < 1 (the case λ = 1 is covered by Theorem 1 with p = 2).

Observe that a nonexistence result for (2) implies the rather strong assertion that all semilinear inequalities with fixed ellipticity constants and L ∞ -bounds for the coefficients have no solutions at infinity. So it should come as no surprise that in order to prove nonexistence of positive solutions of (2) we always have to make a hypothesis on the behavior of f (t) at t = 0. It turns out that close to zero f (t) should be no worse than a power t σ , where σ = σ(λ, N ) tends to 2 * = N/(N -2) when λ → 1, and σ tends to 1 when λ → 0. In addition, we discover that when the ellipticity is too bad (that is, λ is too close to zero depending on N ), we need to impose a condition on f (t) at t = ∞ as well.

Theorem 2 ([2]). Set λ * = N -1 + λ N -1 -λ and suppose that lim inf t→0 t -λ * f (t) > 0.
In addition, assume that

(i) if λ = 1 N -1 , then lim inf t→∞ e at f (t) > 0 for each a > 0; and (ii) if λ < 1 N -1 , then lim inf t→∞ t |λ * | f (t) > 0, where λ * = N -1 + 1/λ N -1 -1/λ .
Then the inequality (2) has no positive weak solution.

The second hypothesis in Theorem 2 is not very strong. It is needed only when λ < 1/(N -1) and allows f (t) to decay to zero when t goes to infinity, but no worse than t -|λ * | .

Theorem 2 is again optimal, in the sense that we can construct a solution of ( 2), provided we take f to be a model nonlinearity which does not satisfy one of the hypotheses of the theorem (see for instance [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF]).

All previous papers on nonexistence for inequalities in non-divergence form concerned the nonlinearity f (t) = t q (with the exception of [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF] where we imposed a more general but still global hypothesis on f ). A list of references is given in [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF]; we only mention here that it was proved by Cutri and Leoni [START_REF] Cutrì | On the Liouville property for fully nonlinear equations[END_REF] that the inequality -P + λ,Λ (D 2 u) ≥ u q has no positive solutions in the whole space R N provided q ∈ (0, λ * ]. It follows in particular from Theorem 2 that this inequality has no solutions even in any exterior domain of R N , for the larger range q ∈ (-∞, λ * ] if λ(N -1) ≥ 1, and q ∈ [λ * , λ * ] if λ(N -1) < 1. Of course, Theorem 2 goes much further, by showing that only the behavior of f (t) close to t = 0 and t = ∞ matters, and by describing with precision the behavior which may be allowed.

Theorems 1 and 2 are very particular cases of Corollary 4.2 in [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF]. The proof of this result is based on a new argument which, in addition to yielding new and optimal results on nonexistence, has several advantages for proving these kinds of Liouville theorems. Above all, it is based entirely on very general maximum principle ideas, which renders it applicable to a great variety of elliptic equations and systems, set in various unbounded domains. We have shown in [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF] how our method trivially extends to systems of elliptic inequalities in exterior domains, still giving optimal results for such systems. We also show in [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF] and in our forthcoming work [START_REF] Armstrong | Singular solutions of fully nonlinear elliptic equations in cones[END_REF] that it yields new nonexistence results in conical domains, and explains the somewhat different phenomena which occur in such domains.

Next, besides its obvious simplicity, the argument makes very apparent the interplay between the scaling of the differential inequality and the scaling of any given subsolution of the differential operator. Optimal results are obtained when this subsolution is taken to be the fundamental solution of the operator. Finally, our method is independent of the nature of the equations considered, in divergence or non-divergence form, or of the nature of their weak solutions, as long as they satisfy a weak comparison principle. It is actually possible to axiomatize the properties of the elliptic operators involved, under which the method can be applied. We refer to [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF] for a discussion; we expect variations of our method to apply to even larger classes of inequalities.

In the next section we describe the proof of Theorem 1, dividing it into three parts. We start by giving a list of its main ingredients, then prove some simple particular cases of the theorem which require only subsets of these ingredients, and finally we expose the full proof.

Proof of Theorem 1

In this section we give the proof of Theorem 1. The proof of Theorem 2 is practically the same, see the end of this section. To fix ideas, in the sequel we assume G ⊂ B 1 (for each r > 0 we denote with B r the ball of radius r) and set Φ p (x) = |x| (p-N )/(p-1) if p = N , and

Φ p (x) = log(3|x|) if p = N .
The basic idea of the proof of Theorem 1 is very simple. The term f (u) on the right side of (1) forces a hypothetical supersolution u of (1) to be small. This is because, as for example in the case f is superlinear near t = 0 and p < N , if u were not small then the right-side would be too big for the left side of (1). On the other hand, by the comparison principle, the fundamental solution provides a lower bound for u(x) for large |x|. This can be seen by "sliding" the fundamental solution Φ p underneath u. These two forces are obviously in conflict, and we would like to understand when this conflict is fatal to the existence of u.

The ingredients of the proof.

The key tool we use in estimating u is the following growth lemma, which is a quantitative version of the strong maximum principle. For an easy proof we refer to [2, Theorem 3.3].

Lemma 3. Assume h ∈ L ∞ (B 3 \ B 1/2 ) is nonnegative, and u ≥ 0 satisfies -∆ p u ≥ h(x) in B 3 \ B 1/2 .
(a) For each A ⊂ B 3 \ B 1/2 there exists a constant c 0 > 0 depending only on N and |A|, such that inf

B 2 \B 1 u ≥ c 0 inf A h 1/(p-1)
.

(b) Suppose in addition that u ≥ kΦ p in B 3 \ B 1/2 for some k > 0. Then we have the estimate

inf B 2 \B 1 (u -kΦ p ) ≥ c 0 inf A h 1/(p-1)
. 2.2. Some particular cases of Theorem 1. Let us first assume p < N . In order to simplify the following proofs we will strengthen (i) and assume that ( 5)

lim t→0 t -p * f (t) = ∞.
1. First we are going to show that only Lemmas 3 and 4 are sufficient to prove that (1) has no positive solutions provided (6) lim inf t→∞ f (t) t p-1 > 0. Proof. For r ≥ 2 we define the rescaled function u r (x) := u(rx) and observe that (1) may be written in terms of u r as

-∆ p u r ≥ r p f (u r ) in R N \ B 1/r ⊃ B 3 \ B 1/2 . Since m(r) = inf B 2 \B 1 u r we immediately obtain -∆ p u r ≥ r p min t∈[m(r),∞) f (t) χ B 2 \B 1 in B 3 \ B 1/2 ,
where χ Z denotes the characteristic function of a set Z ⊂ R N . Applying Lemma 3 (a) with

A = B 2 \ B 1 we deduce that (7) m(r) p-1 ≥ c 0 r p min t∈[m(r),∞) f (t) ,
for some c 0 > 0 independent of r. By ( 6) the minimum in the right-hand side of ( 7) is attained, say at a point m(r) ∈ [m(r), ∞). So ( 7) implies ( 8) m(r) 1-p f ( m(r)) ≤ c -1 0 r -p , Sending r → ∞ in [START_REF] Gidas | Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations[END_REF] we see that [START_REF] Cutrì | On the Liouville property for fully nonlinear equations[END_REF], the continuity of f and f (t) > 0 for t > 0 imply m(r) → 0 as r → ∞. Hence [START_REF] Bidaut-Véron | Local and global behavior of solutions of quasilinear equations of Emden-Fowler type[END_REF] implies that for sufficiently large r we have [START_REF] Kondratiev | Positive solutions to semi-linear and quasilinear elliptic equations on unbounded domains[END_REF] f ( m(r)) ≥ C r m(r) p * , where C r → ∞ as r → ∞. Recalling the definition of p * (which in particular implies that p -1 -p * = -p(p -1)/(N -p)) and combining ( 9) with ( 8) and the lower bound on m(r) in Lemma 4 yields

(10) C r r -p ≤ C r m(r) p * -p+1 ≤ C r m(r) p * -p+1 ≤ c -1 0 r -p , which is a contradiction, since C r → ∞ as r → ∞.
2. We now observe that adding Lemma 5 to the above argument permits us to relax the hypothesis ( 6) to the following one [START_REF] Ni | Nonexistence theorems for quasilinear partial differential equations[END_REF] lim inf t→∞ f (t) > 0.

Proof. By Lemma 5 the left-hand side of (7) is bounded above, so the minimum in the righthand side tends to zero as r → ∞. Again by [START_REF] Ni | Nonexistence theorems for quasilinear partial differential equations[END_REF] and f (t) > 0 for t > 0 this minimum is attained and we get m(r) → 0 as r → ∞, so ( 9)-( 10) hold, yielding the same contradiction as above.

Suppose now that p ≥ N . Note we obtained (7) by using only Lemma 3, so this inequality is independent of the value of p > 1. In the case p ≥ N Lemma 5 implies that m(r) is bounded below by a positive constant for large r. Hence if we assume [START_REF] Ni | Nonexistence theorems for quasilinear partial differential equations[END_REF] we see that the minimum in the right-hand side of ( 7) is bounded below by a positive constant, by the continuity and positivity of f . Then [START_REF] Lorenzo | A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities[END_REF] implies that m(r) ≥ cr p/(p-1) which is a contradiction with the upper bound in Lemma 4, for all sufficiently large r. We arrive at the same contradiction if we use only Lemma 3 and Lemma 4, but assume in addition to [START_REF] Ni | Nonexistence theorems for quasilinear partial differential equations[END_REF] that lim inf t→0 f (t)

t p-1 > 0.
We remark that in the simple proofs above we used only the trivial particular case of Lemma 3 (a) with

A = B 2 \ B 1 .
Let us now move to the hard part of the proof of Theorem 1, the removal of any condition on f at infinity if p < N , resp. allowing f to decay at infinity if p ≥ N . It is here that we need the very weak Harnack inequality (Lemma 7) as well as the growth lemma (Lemma 3) in its full strength.

2.3. Proof of Theorem 1. Let us suppose that u > 0 is a supersolution of (1) in R N \ B 1 .
We consider first the case that p < N .

For r ≥ 2 we again define the rescaled function u r (x) := u(rx) and recall that (1) may be written in terms of u r as

-∆ p u r ≥ r p f (u r ) in R N \ B 1/r ⊃ B 3 \ B 1/2 .

Denote

A r := {x ∈ B 2 \ B 1 : m(r) ≤ u r (x) ≤ Cm(r)}, where C = C(N, 1 2 ) > 1 is as in Lemma 7, which yields that

|A r | ≥ (1/2)|B 2 \ B 1 | = c(N ) > 0.
Lemma 3 (a) with A = A r and h(x) := r p f (u r (x))χ Ar (x) yields the estimate m(r) p-1 ≥ c 0 r p min f (t) : m(r) ≤ t ≤ Cm(r) for each r ≥ 2, [START_REF] Serrin | Isolated singularities of solutions of quasi-linear equations[END_REF] and some c 0 > 0 independent of r. According to Lemma 5, the quantity m(r) is bounded above for r > 2, and so we have [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF] min [m(r), Cm(r)] f ≤ Cr -p m(r) p-1 ≤ Cr -p → 0 as r → ∞.

We deduce from m(r) ≤ C that the interval over which this minimum is taken is bounded above. Since f is continuous and positive on (0, ∞), m(r n ) ≥ c > 0 for some sequence r n → ∞ is clearly in contradiction with [START_REF] Serrin | Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities[END_REF]. Hence [START_REF] Trudinger | On Harnack type inequalities and their application to quasilinear elliptic equations[END_REF] m(r) → 0 as r → ∞.

We remark that if instead of (i) we assumed the stronger hypothesis [START_REF] Bidaut-Véron | Local and global behavior of solutions of quasilinear equations of Emden-Fowler type[END_REF], at this stage we can deduce the inequalities ( 8)- [START_REF] Mitidieri | A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities[END_REF] (with m(r) ∈ [m(r), Cm(r)]) and hence a contradiction. Let us continue with the proof of the Theorem assuming only (i). This assumption and (12) imply that for r > 2 sufficiently large, [START_REF] Véron | Singularities of solutions of second order quasilinear equations[END_REF] (m(r)) p-1 ≥ cr p m(r) p * .

Since p -1 -p * = -p(p -1)/(N -p), we can rearrange this inequality as ( 16) m(r) ≤ Cr -α * for every sufficiently large r ≥ 2.

where, as before, α * = (N -p)/(p -1). Note ( 16) and the fact that Φ p is -α * -homogeneous imply that the quantity ρ(r) is bounded above for r > 2.

We recall that, according to Lemma 4, for some c > 0 we also have ( 17) m(r) ≥ cr -α * for every r > 2.

Observe that by Lemma 4 ρ(r) is nondecreasing in r, that is ρ(r) = inf R N \Br u Φ p . We are going to apply the second part of Lemma 3 to the function

v r (x) := r α * u(rx), which satisfies v r ≥ ρ(r/2) r α * Φ p (rx) = ρ(r/2) Φ p (x) on R N \ B 1/2 . By (17) we have v r ≥ c > 0 on B 2 \ B 1
, where c > 0 does not depend on r. It also follows from the above argument (inequalities ( 12)-( 17)) that

-∆ p u ≥ f (u) u p * u p * ≥ cu p * on 1 r A r .
By the scaling invariance and the choice of p * and α * (recall the discussion following (3)) we see that v r then satisfies

-∆ p v r ≥ cv p * r on A r , hence -∆ p v r ≥ cχ Ar on B 3 \ B 1/2 . By Lemma 3 (b) this implies v r ≥ ρ(r/2)Φ p + c ≥ (ρ(r/2) + c)Φ p on B 2 \ B 1 ,
where c > 0 does not depend on r. The definition of ρ(r) and the last inequality yield ρ(r) ≥ ρ(r/2) + c 1 for each r > 2, and therefore lim r→∞ ρ(r) = ∞, which contradicts our inequality (16).

We next consider the case p ≥ N . As before, we arrive at the inequality [START_REF] Serrin | Isolated singularities of solutions of quasi-linear equations[END_REF]. Now applying Lemma 5, we see that m(r) is bounded away from 0, in contrast to the previous case p < N above. Since f (t) is continuous and positive for t > 0 it follows from the first inequality in (13) that ( 18) m(r) → ∞ as r → ∞.

We split the remainder of the argument into the cases p = N and p > N .

In the case p = N , we deduce from ( 12) and (ii) that for every a > 0 there exist ε = ε(a) > 0 small enough and r 0 > 1 large enough that m(r) p-1 e a Cm(r) ≥ cεr p . for r > r 0 . This inequality trivially implies that for each A > 0 there exists r 1 > r 0 large enough that m(r) ≥ A log r for r > r 1 , which contradicts the upper bound in Lemma 4.

Finally, in the case that p > N , from ( 12) and (iii) we obtain m(r) ≥ cr (p-N )/(p-1) for some constant c > 0 and large enough r. Lemma 4 gives the reverse inequality, so that for large r we have the two-sided bound (19) cr -α * ≤ m(r) ≤ Cr -α * , that is, 0 < c ≤ ρ(r) ≤ C for all large r. We set again v r (x) := r α * u(rx), note that 0 < c ≤ v r ≤ C on the set A r , and argue exactly like in the case p < N to deduce ρ(r) → +∞ as r → ∞, in contradiction to (19). The proof is complete.

Remark 8. In the case when p = N the function (Φ p ) τ is a solution of -∆ p u = u q in R N \ {0}, for a suitable chosen power τ , provided (i) or (iii) is not satisfied. By using a cutoff like argument it is possible to modify these functions to obtain solutions of -∆ p u ≥ u q in R N (see for instance the argument on page 11 in [START_REF] Armstrong | Liouville results for fully nonlinear elliptic equations with power growth nonlinearities[END_REF]). If p = N it is also easy to obtain solutions of -∆ p u = e au , a > 0, in exterior domains. For instance u(x) := 2 a (log |x| + log (log |x|)) is a positive solution of the equation -∆u = 2 a e -au in R 2 \ B 3 . Note that for p = N every positive solution of -∆ p u ≥ 0 in R N \ {0} is constant (by Lemma 5 and Proposition 6).

These remarks attest to the sharpness of Theorem 1.

Finally, we observe that the same argument leads to the proof of Theorem 2. We only need to replace the fundamental solution |x| (N -p)/(p-1) by |x| λ -1 (N -1)-1 , and its opposite -|x| (N -p)/(p-1) by -|x| λ(N -1)-1 (of course some care is needed about the different homogeneities of the functions |x| λ -1 (N -1)-1 , -|x| λ(N -1)-1 , which both solve P + λ,Λ (u) = 0 in the punctured space R N \ {0}). Theorem 2 is also sharp, as adequate powers of these functions show.

Acknowledgement. The first author was supported in part by NSF Grant DMS-1004645.

The fundamental solution Φ p and its opposite -Φ p give bounds on the decay (or growth) of any positive p-superharmonic function in an exterior domain. This is summarized in the next two lemmas which cause the hypothesis of Theorem 1 to break into the different cases it does. These lemmas are known, though not so often used; we refer to [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF]Lemma 3.7] for simple proofs based on the comparison principle.

For each r ≥ 1, we have Φ p > 0 in R N \ B r , and so we may define (4) m(r) := inf

In particular m(r

Lemma 4 is proved by sliding underneath u functions of the type AΦ p + B, for suitably chosen constants A > 0, B ∈ R.

To prove Lemma 5 we compare u from below with functions of the type A(-Φ p ) + B, for suitable A > 0, B ∈ R.

Even if we will not use it here, for completeness we recall the following simple property, which may be combined with the above lemma. Proposition 6. Assume that u ≥ 0 is p-superharmonic in R N or that p ≤ N and u ≥ 0 is p-superharmonic in R N \ {0}. Then m(r) is nonincreasing in r > 0.

In order to get the optimal local behavior of the nonlinearity f , we will also need the following measure theoretic estimate. Lemma 7. For every 0 < ν < 1, there exists a constant C = C(N, ν) > 1 such that for any positive p-superharmonic function u in B 3 \ B1/2 and any

We remark that Lemma 7 is easily seen to be weaker than the weak Harnack inequality proved for example in [START_REF] Serrin | Isolated singularities of solutions of quasi-linear equations[END_REF][START_REF] Trudinger | On Harnack type inequalities and their application to quasilinear elliptic equations[END_REF]. See also [START_REF] Armstrong | Nonexistence of positive supersolutions of elliptic equations via the maximum principle[END_REF]Remark 3.6].

We will next show how these ingredients can be combined to yield nonexistence results. Before giving the full proof of Theorem 1 which requires all lemmas above, we are going to prove several particular cases which are particularly simple (but still more general than what is usually encountered in the literature) and which need only a subset of these lemmas. We do this in order to, on one hand, better highlight the main points in the proofs, and on the other hand, facilitate eventual extensions of our method to situations in which not all of the above lemmas are available.