Philippe Laurenc ¸ot

Christoph Walker
email: walker@ifam.uni-hannover.de

A FOURTH-ORDER MODEL FOR MEMS WITH CLAMPED BOUNDARY CONDITIONS

Keywords: Mathematics Subject Classification. 35J40, 35J75, 35J91, 35L75, 35K35, 74F15 MEMS, fourth-order diffusion, beam equation, positivity, quenching, touchdown

The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems (MEMS) and includes a positive voltage parameter λ. It is shown that there is a threshold value λ * > 0 of the voltage parameter such that no radially symmetric stationary solution exists for λ > λ * , while at least two such solutions exist for λ ∈ (0, λ * ). Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when λ > λ * .

INTRODUCTION

Electrostatically actuated microelectromechanical systems (MEMS) are microscopic devices which combine mechanical and electrostatic effects. A typical MEMS device is made of a rigid conducting ground plate above which a clamped deformable membrane coated with a thin conducting film is suspended. Application of a voltage difference induces a Coulomb force which, in turn, generates a displacement of the membrane. An ubiquitous feature of such devices is, that when the applied voltage exceeds a certain threshold value, the membrane might collapse (or touch down) on the ground plate. Controlling the occurrence of this phenomenon -usually referred to as the "pull-in" instability -is of utmost practical importance in the design of such devices either to set up optimal operating conditions or to avoid device damaging. Mathematical models have been derived to describe MEMS devices which lead to free boundary problems due to the deformable membrane [START_REF] Pelesko | Modeling MEMS and NEMS[END_REF]. Since these models are difficult to analyze mathematically (though recent contributions can be found in [START_REF] Cimatti | A free boundary problem in the theory of electrically actuated microdevices[END_REF][START_REF] Escher | A parabolic free boundary problem modeling electrostatic MEMS[END_REF][START_REF] Laurenc | A stationary free boundary problem modeling electrostatic MEMS[END_REF]), one often takes advantage of the small aspect ratio of the devices to reduce the free boundary problem to a single equation for the displacement, see [START_REF] Pelesko | Modeling MEMS and NEMS[END_REF]. More precisely, the small aspect ratio model describing the dynamics of the displacement u = u(t, x) of the membrane Ω ⊂ R d reads

γ 2 ∂ 2 t u + ∂ t u + B∆ 2 u -T ∆u = - λ (1 + u) 2 , t > 0 , x ∈ Ω , (1.1)
u = ∂ ν u = 0 , t > 0 , x ∈ ∂Ω , (1.2)
u(0, •) = u 0 , ∂ t u(0, •) = u 1 , x ∈ Ω . (1.3)
Here, γ 2 ∂ 2 t u and ∂ t u account, respectively, for inertia and damping effects, B∆ 2 u and -T ∆u are due to bending and stretching of the membrane, while -λ(1 + u) -2 reflects the action of the electrostatic forces in the small aspect ratio limit. The parameter λ is proportional to the square of the applied voltage. Observe that the right-hand side of (1.1) features a singularity when u = -1, which corresponds to the touchdown phenomenon already mentioned above. Since the strength of the singular reaction term is tuned by the parameter λ, it is not surprising that the latter governs the existence of stationary solutions, that is, solutions to

B∆ 2 u -T ∆u = - λ (1 + u) 2 , x ∈ Ω , (1.4)
u = ∂ ν u = 0 , x ∈ ∂Ω . (1.5)
When bending is neglected, that is, when B = 0, this problem reduces to a second-order elliptic equation that has been studied extensively in the recent past, see e.g. the monograph [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] and the references therein. As expected from the physics there is a critical value λ * > 0 such that no stationary solution exists if λ > λ * and at least one stationary solution exists for λ ∈ (0, λ * ).

Let us emphasize that the comparison principle is available in this case and turns out to be a key tool for the analysis. Less attention has been dedicated to (1.4)-(1.5) with B > 0, one reason might be the lack of a maximum principle in general for the clamped boundary conditions (1.5) (also called Dirichlet boundary conditions), see the monograph [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF] for a detailed discussion of positivity properties of higher-order operators. Recall that the situation is completely different if the clamped boundary conditions (1.2) are replaced with pinned (or Navier) boundary conditions u = ∆u = 0 on ∂Ω , (1.6) since in this case the maximum principle holds in arbitrary domains [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF][START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]. This allows one in particular to show similar results for the fourth-order problem (1.4), (1.6) as outlined above for the second-order case corresponding to B = 0. We refer to [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF][START_REF] Lin | Nonlinear non-local elliptic equation modelling electrostatic actuation[END_REF] for details.

Returning to the case of clamped boundary conditions, when B > 0 existence of solutions to (1.4)- (1.5) for small values of λ has been established in [START_REF] Lin | Nonlinear non-local elliptic equation modelling electrostatic actuation[END_REF] for an arbitrary domain Ω. This is the only result we are aware of for a general domain. In the particular case when Ω equals the unit ball B 1 , Boggio [START_REF] Boggio | Sulle funzioni di Green d'ordine m[END_REF] has uncovered the availability of the maximum principle for the operator B∆ 2 with boundary conditions (1.5) by showing that the corresponding Green function is positive. This fact has been used in [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] to describe more precisely the set of solutions to (1.4)-(1.5) with T = 0. It has actually been shown in [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF]Chapter 11] that there is a critical threshold value λ * > 0 such that no solution exists for λ > λ * and a solution exists for λ ∈ (0, λ * ].

Very recently we were able to extend Boggio's maximum principle to the operator B∆ 2 -T ∆ with T > 0 and boundary conditions (1.5) in the class of radially symmetric functions in B 1 [START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF]. Taking advantage of this property not only allows us to extend the results of [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] to include T > 0 (for d = 1, 2), but also to show that for each voltage value λ ∈ (0, λ * ) there are at least two (radially symmetric) solutions to (1.4)- (1.5), thereby answering a question raised in [10, p. 268]. A summary of our results on radially symmetric solutions to (1.4)-(1.5) is stated in the following theorem.

Theorem 1.1. Let d = 1, 2, Ω = B 1 , B > 0, and T ≥ 0. There exists λ * > 0 such that there is no radially symmetric solution to (1.4)-(1.5) for λ > λ * . Moreover, there is a continuous curve (Λ(s), U(s)), s ∈ [0, ∞) in R × C 4 ( B1 ) such that U(s) is for each s ∈ [0, ∞) a radially symmetric solution to (1.4)-(1.5) with λ = Λ(s). Moreover, (Λ(0), U(0)) = (0, 0) and (Λ(s), U(s)) → (0, ω) as s → ∞, where ω is an explicitly given radially symmetric function with ω(0) = -1. Finally, there is s * > 0 such that Λ is an increasing function from [0, s * ] onto [0, λ * ] and Λ is decreasing in a right-neighborhood of s * .

Remark 1.2. Theorem 1.1 guarantees that for each λ ∈ (0, λ * ), there are at least two (radially symmetric) solutions to (1.4)-(1.5). More precisely, for each λ ∈ (0, λ * ) there are at least two values 0 < s

1 < s * < s 2 with Λ(s j ) = λ for j = 1, 2 and U(s 2 ) ≤ U(s 1 ) in B 1 with U(s 2 ) = U(s 1 ).
Let us mention here that the construction of the curve (Λ(s), U(s)), s ∈ [0, s * ] follows the lines of [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF]Chapter 11], where a similar result is proved when T = 0. There are, however, some technical difficulties to be overcome. Nevertheless, we emphasize that the main contributions of Theorem 1.1 are the extension of the curve past (Λ(s * ), U(s * )) and the identification of its end point ω as s → ∞. Interestingly, the end point ω is given as a solution of a boundary value problem in B 1 \ {0} which can be computed explicitly (see Theorem 2.20 below); a plot of ω is shown in Figure 1. The qualitative behavior of ω is the same for d = 1 and d = 2.

For the case of pinned boundary conditions (1.6) it has been shown in [START_REF] Guo | On a fourth order nonlinear elliptic equation with negative exponent[END_REF]Theorem 1.2] with the help of the Mountain Pass Principle that there are at least two solutions for λ ∈ (0, λ * ). The limit as λ → 0 of the minimum of the solutions constructed with the Mountain Pass Principle is proved to be -1. However, the precise profile as λ → 0 is not identified therein.

The proof of Theorem 1.1 is performed in Section 2. Therein we give a more detailed characterization of the set of radially symmetric stationary solutions. Actually, the implicit function theorem provides a branch A 0 of radially symmetric solutions (λ, u) to (1.4)-(1.5) emanating from (0, 0). We then use the bifurcation theory of [START_REF] Buffoni | The sub-harmonic bifurcation of Stokes waves[END_REF] for real analytic functions to extend A 0 to a global curve A (see Theorem 2.5 below). Next we show that A 0 coincides with the branch of stable radially symmetric stationary solutions (see Corollary 2.16). To achieve this result, the maximum principle obtained in [START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF] is essential as was Boggio's maximum principle in [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] for the case T = 0. The outcome of this analysis is that there is a threshold value λ * > 0 such that there is no radially symmetric stationary solution for λ > λ * , while for any λ ∈ (0, λ * ) there is a unique stationary solution u λ such that (λ, u λ ) ∈ A 0 . We then show that for λ = λ * there is also a radially symmetric stationary solution u λ * , which guarantees on the one hand that A 0 = A and on the other hand that we may apply the result of [START_REF] Michael | Bifurcation, perturbation of simple eigenvalues and linearized stability[END_REF] to extend the branch A 0 to the "right" of (λ * , u λ * ) (see Theorem 2.18). The final step is to show that A connects (λ, u) = (0, 0) to the end point (0, ω) and to identify the latter (see Theorem 2.20). As a consequence, the continuous curve A passes through (λ, u) = (λ * , u λ * ), which implies Remark 1.2.

We shall also investigate local and global well-posedness of the dynamic problem (1.1)- (1.3). It is worth pointing out that the maximum principle, which is at the heart of the proof of Theorem 1.1 and the main reason to restrict the analysis to B 1 , is no longer valid for the timedependent problem (1.1)- (1.3). In order to construct solutions to the evolution problem, we therefore have to employ an alternative method which does not rely on maximum principles. Our approach is based on semigroup theory and is not specific to B 1 . We thus consider an arbitrary domain Ω ⊂ R d , d = 1, 2, in the following and begin with the hyperbolic problem which has not received much attention so far.

Theorem 1.3. Let Ω ⊂ R d be an arbitrary bounded smooth domain for d = 1, 2 and B > 0, T ≥ 0. Let λ > 0 and κ ∈ (0, 1). Let (u 0 , u 1 ) ∈ H 4 (Ω) × H 2 (Ω) be such that u 0 ≥ -1 + κ in Ω and such that u 0 and u 1 both satisfy the boundary conditions (1.2). Then the following hold:

(i) There are τ m > 0 and a unique maximal solution u to (1.1)-(1.3) with regularity

u ∈ C([0, τ m ), H 2 (Ω)) ∩ C 1 ([0, τ m ), L 2 (Ω)) , ∂ k t u ∈ L 1 (0, τ ; H 4-2k (Ω))
for k = 0, 1, 2 and τ ∈ (0, τ m ).

(ii) If τ m < ∞, then lim inf t→τm min Ω u(t) = -1 . (iii) There are λ 1 (κ) > 0 and r(κ) > 0 such that τ m = ∞ provided that λ ≤ λ 1 (κ) and (u 0 , u 1 ) H 2 ×L 2 ≤ r(κ). In this case, u ∈ L ∞ (0, ∞; H 2 (Ω)) and inf (0,∞)×Ω u > -1 .
(iv) If Ω = B 1 and both u 0 and u 1 are radially symmetric, then so is u(t) for each t ∈ [0, τ m ).

Similar results have been established in [START_REF] Kavallaris | A hyperbolic non-local problem modelling MEMS technology[END_REF] for d = 1 and B = 0 (without the damping term ∂ t u) and in [START_REF] Guo | Dynamical solutions of singular wave equations modeling electrostatic MEMS[END_REF] for the pinned boundary conditions (1.6) and d ∈ {1, 2, 3}. Let us point out that the semigroup approach allows us to obtain strong solutions instead of weak solutions as in [START_REF] Guo | Dynamical solutions of singular wave equations modeling electrostatic MEMS[END_REF][START_REF] Kavallaris | A hyperbolic non-local problem modelling MEMS technology[END_REF], see Section 3.1 for the proof of Theorem 1.3.

In the damping dominated limit γ ≪ 1 when viscous forces dominate over inertial forces, (1.1)-(1.3) reduces to the parabolic problem

∂ t u + B∆ 2 u -T ∆u = - λ (1 + u) 2 , t > 0 , x ∈ Ω , (1.7
)

u = ∂ ν u = 0 , t > 0 , x ∈ ∂Ω , (1.8)
u(0, •) = u 0 , x ∈ Ω . (1.9)
To the best of our knowledge, this problem has not been investigated so far. For this reason, we include a result on its well-posedness though local existence of solutions is a rather classical argument. To obtain global solutions for small values of λ, we consider only regular initial values in the next theorem for the sake of simplicity.

Theorem 1.4. Let Ω ⊂ R d be an arbitrary bounded smooth domain for d = 1, 2 and B > 0, T ≥ 0. Let λ > 0 and κ ∈ (0, 1). Let u 0 ∈ H 2 (Ω) be such that u 0 ≥ -1 + κ in Ω and u 0 satisfies the boundary conditions (1.8). Then the following hold: (i) There are τ m > 0 and a unique maximal solution u to (1.7)-(1.9) with regularity

u ∈ C([0, τ m ), H 2 (Ω)) ∩ C((0, τ m ), H 4 (Ω)) ∩ C 1 ((0, τ m ), L 2 (Ω)) . (ii) If τ m < ∞, then lim inf t→τm min Ω u(t) = -1 . (iii) There is λ 1 (κ) > 0 such that τ m = ∞ provided that λ ≤ λ 1 (κ) and u 0 H 2 ≤ κ -1 . In this case, u ∈ L ∞ (0, ∞; H 2 (Ω)) and inf (0,∞)×Ω u > -1 .
(iv) If Ω = B 1 and u 0 is radially symmetric, then so is u(t) for each t ∈ [0, τ m ).

The proof of Theorem 1.4 follows the same lines as that of Theorem 1.3 and is to be found in Section 3.2.

On physical grounds it is expected that solutions to the dynamic problems (1.1)-(1.3) or (1.7)-(1.9) touch down (i.e. u = -1) in finite time and thus do not exist globally if the voltage value λ exceeds the critical pull-in voltage above which no stationary solution exists. This is true for the second-order parabolic case, see [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] for instance, but seems to be an open problem both for the hyperbolic equation (1.1)-(1.3) as well as for the parabolic equation (1.7)-(1.9). Actually, even the weaker result of the occurrence of touchdown in finite time for large values of λ has not yet been proven apparently, though observed numerically in [START_REF] Lindsay | Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor[END_REF] for (1.7)-(1.9) with T = 0 and shown in [START_REF] Kavallaris | A hyperbolic non-local problem modelling MEMS technology[END_REF] in the absence of bending (B = 0). The next result is a step in that direction when Ω is the unit ball B 1 of R d . Proposition 1.5. Let Ω = B 1 with d ∈ {1, 2}, λ > 0, B > 0, T ≥ 0, and let (u 0 , u 1 ) ∈ H 4 (B 1 ) × H 2 (B 1 ) be such that u 0 > -1 in B 1 and both u 0 and u 1 satisfy the boundary conditions (1.2). Let u be the maximal solution to either (1.1)-(1.3) with initial condition (u 0 , u 1 ) enjoying the properties listed in Theorem 1.3 or (1.7)-(1.9) with initial condition u 0 enjoying the properties listed in Theorem 1.4. Let τ m be its maximal existence time. If λ is sufficiently large (see (4.2) below for a quantitative lower bound), then τ m < ∞.

It is worth pointing out that the outcome of Proposition 1.5 complies with the numerical simulations of (1.7)-(1.9) performed in [START_REF] Lindsay | Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor[END_REF] in B 1 and showing the occurrence of finite time touchdown. The proof of Proposition 1.5 is given in Section 4.1 and relies on the eigenfunction method.

Owing to the study carried out in Section 2, we are able to refine this result in the radially symmetric setting and show that the touchdown behavior indeed starts exactly above the threshold value λ * defined in Theorem 1.1.

Proposition 1.6. Assume Ω = B 1 with d ∈ {1, 2} and let (u 0 , u 1 ) be radially symmetric initial conditions satisfying the requirements of Theorem 1.3 if γ > 0 or Theorem 1.4 if γ = 0. Then, if λ > λ * , the corresponding maximal solution to (1.1)-(1.3) or (1.7)-(1.9) on [0, τ m ) does not exist globally, that is, τ m < ∞.

The proof of Proposition 1.6 is performed in Section 4.2 and also relies on the eigenfunction method, but with a more accurate choice than in the proof of Proposition 1.5 as already noticed in [START_REF] Guo | Dynamical solutions of singular wave equations modeling electrostatic MEMS[END_REF].

Let us conclude the introduction with some remarks on the qualitative behavior of solutions to the evolution problem in the ball B 1 . Proposition 1.5 and Proposition 1.6 show the occurrence of a finite time singularity, but do not provide information about the precise behavior near touchdown time. According to the numerical simulations performed in [START_REF] Lindsay | Multiple quenching solutions of a fourth order parabolic PDE with a singular nonlinearity modeling a MEMS capacitor[END_REF], the fourth-order term has a strong influence on the way solutions touch down in finite time as this might take place on a circle (for d = 2). This markedly contrasts the second-order case, where touchdown occurs only at the single point x = 0, see [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF]Theorem 8.3.4].

When a solution to (1.1)-(1.3) does not touch down in finite time, then it exists globally in time and might even be bounded away from -1 as well as be bounded in H 2 according to Theorem 1.3 (if γ > 0) and Theorem 1.4 (if γ = 0). A natural next step to understand its dynamics is to investigate its large time behavior. While this seems to be an open problem for a general domain Ω, the analysis performed in Section 2 for Ω = B 1 in the radially symmetric setting paves the way for a better understanding of this issue. On the one hand, Proposition 2.15 below entails that one may apply the principle of linearized stability to show that U(s) is locally asymptotically stable when s ∈ [0, s * ). On the other hand, it might be possible to use the Łojasiewicz-Simon inequality as in [START_REF] Guo | Dynamical solutions of singular wave equations modeling electrostatic MEMS[END_REF] to establish convergence to a single steady-state.

RADIALLY SYMMETRIC STATIONARY SOLUTIONS

In the following, if S(Ω) is a space of functions defined on Ω, we write S D (Ω) for the subspace of S(Ω) consisting of functions u satisfying the Dirichlet boundary conditions (1.5), if meaningful. If Ω is the unit ball B 1 of R d , then S r (B 1 ) stands for the subspace of S(B 1 ) consisting of radially symmetric functions. Clearly, S D,r (B 1 ) := S D (B 1 ) ∩ S r (B 1 ). If S(Ω) is a normed vector space, then • S stands for its norm. For p ∈ [1, ∞] we denote the norm of L p (Ω) simply by • p .

Recall that the stationary solutions of (1.1)-(1.2) satisfy

B∆ 2 u -T ∆u = -λg(u) in B 1 , (2.1)
u = ∂ ν u = 0 on ∂B 1 , (2.2)
where g(ξ) := (1 + ξ) -2 for ξ > -1.

Definition 2.1. A radially symmetric classical solution u (with parameter λ) of the boundary value problem

(2.1)-(2.2) is a radially symmetric function u ∈ C 4 r (B 1 ) ∩ C 2 r ( B1 ) satisfying u > -1 in B1 and solving (2.1)-(2.2) in the classical sense.
We denote the set of all radially symmetric classical solutions with parameter λ to the bound-

ary value problem (2.1)-(2.2) by S λ r . Similarly, a radially symmetric function u ∈ C 4 r (B 1 ) ∩ C 2 r ( B1 ) satisfying u > -1 in B1 is a classical subsolution of (2.1)-(2.2) (with parameter λ), if it satisfies B∆ 2 u -T ∆u ≤ -λg(u) in B 1 and (2.2) on ∂B 1 .
We introduce the operator

Au := B∆ 2 u -T ∆u , u ∈ H 4 D,r (B 1
) , and recall the following well-known properties:

Lemma 2.2. A ∈ L(H 4 D,r (B 1 ), L 2,r (B 1 )) is invertible with A -1 ∈ L(L 2,r (B 1 ), H 4 D,r (B 1 )) ∩ L(C α D,r ( B1 ), C 4+α D,r ( B1 )) for each α ∈ (0, 1). Moreover, there are m 1 > 0 and φ 1 ∈ C 4 D,r ( B1 ) with φ 1 > 0 in B 1 , φ 1 1 = 1, and Aφ 1 = m 1 φ 1 .
Proof. The invertibility of A and the regularity properties of A -1 are consequences of [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]Theorem 2.15,Theorem 2.19,Theorem 2.20]. That there is a positive eigenvalue with a positive eigenfunction follows from [START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF]Theorem 4.7].

We further define

λ * := sup λ > 0 : S λ r is non-empty ∈ [0, ∞] , (2.3)
and first derive some elementary properties of S λ r . Lemma 2.3. The following hold:

(i) S 0 r = {0} and S λ r ⊂ C 4 D,r ( B1 ) for λ > 0; (ii) if λ > 0 and u ∈ S λ r , then -1 < u ≤ 0 in B1 ; (iii) the threshold value λ * defined in (2.3) is finite. Proof. The first statement of (i) readily follows from Lemma 2.2. If u ∈ S λ r , then g(u) belongs to C 2 ( B1 ) since g is smooth in (-1, ∞) and u > -1 in B1 . Thus u ∈ C 4 D,r ( B1 ) by Lemma 2.2. Moreover, -1 < u ≤ 0 in B1 by [16, Theorem 1.4] since λg(u) ≥ 0 (see also Lemma 2.7 below). Consequently, testing (2.1)-(2.2) by φ 1 > 0 introduced in Lemma 2.2 yields -m 1 B 1 φ 1 dx ≤ m 1 B 1 φ 1 u dx = -λ B 1 φ 1 g(u) dx ≤ -λ B 1 φ 1 dx , whence λ ≤ m 1 . Therefore, λ * ≤ m 1 < ∞.
In fact, one can show that λ * < m 1 . Indeed, assume λ * = m 1 for contradiction so that there are sequences λ n → m 1 and u n ∈ S λn r . Then, the above computation actually yields

m 1 -λ n ≥ λ n B 1 φ 1 (g(u n ) -1) dx ≥ 0 .
Since also

B 1 φ 1 (g(u n ) -1) dx = B 1 φ 1 |u n |(u n + 2) (1 + u n ) 2 dx ≥ B 1 φ 1 |u n | dx , we conclude that lim n→∞ λ n B 1 φ 1 g(u n ) dx = m 1 and lim n→∞ B 1 φ 1 |u n | dx = 0 .
From u n ∈ S λn r we obtain

m 1 B 1 φ 1 u n dx = -λ n B 1 φ 1 g(u n ) dx ,
and letting n → ∞ and using the previous limits imply that m 1 = 0, which contradicts Lemma 2.2.

Remark 2.4. Observe that the computation in the proof of Lemma 2.3 excludes also the existence of non-radially symmetric solutions to (1.4)-(1.5) for λ > m 1 . An interesting question is whether there are non-radially symmetric solutions for λ ∈ (λ * , m 1 ). This is not the case when T = 0 as it is shown in [START_REF] Berchio | Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems[END_REF] that all solutions to (1.4)-(1.5) are radially symmetric.

2.1.

A continuous curve of stationary solutions. In this subsection we invoke the global bifurcation theory of [5, Section 2.1] for real analytic functions to establish the existence of a global curve of radially symmetric stationary solutions. This tool has also been used in [10, Section 6.2] for the second-order case (that is, B = 0).

Since O := {u ∈ C 1 D,r ( B1 ) : |u| < 1 in B1 } (2.4) is open in C 1 D,r (
B1 ), the mapping

F : R × O → C 1 D,r ( B1 ) , (λ, u) → u + λA -1 g(u) , (2.5)
is well-defined according to Lemma 2.2 and real analytic. Observe that u ∈ S λ r if (λ, u) ∈ R×O with F (λ, u) = 0, the bound |u| < 1 following from Lemma 2.3. Clearly, F (0, 0) = 0 and the partial Fréchet derivative F u (0, 0) equals the identity in C 1 D,r ( B1 ). Thus, by the implicit function theorem, the zeros of F near (0, 0) are given by a real analytic curve (λ, V (λ)) with V (0) = 0. Moreover, there exists λ 0 ∈ (0, ∞], which is maximal with respect to the existence of a real analytic function

V : [0, λ 0 ) → C 1 D,r ( B1 ) for which F (λ, V (λ)) = 0 and F u (λ, V (λ)) ∈ L(C 1 D,r ( B1 )) is boundedly invertible for each λ ∈ [0, λ 0 ). Consequently, the set S := {(λ, u) ∈ (0, ∞) × O : F (λ, u) = 0 and F u (λ, u) ∈ L(C 1 D,r ( B1 )
) is boundedly invertible} is non-empty as it contains the maximal arc-connected subset

A 0 := {(λ, V (λ)) : λ ∈ (0, λ 0 )} . (2.6)
Note that λ 0 and V are unique and necessarily λ 0 is finite since it belongs to (0, λ * ]. We have thus verified assumption (C1) from [5, Section 2.1]. For (C2) therein we may argue as in [10, p.128] that this assumption merely serves to show in the proof of [START_REF] Buffoni | The sub-harmonic bifurcation of Stokes waves[END_REF]Theorem 2.3 (iii)] that S is open in its closure S and can thus be replaced by the stronger one that (0, [START_REF] Rudin | Functional analysis[END_REF]Theorem 4.25], the partial Fréchet derivative

∞) × O is open in R × C 1 D,r ( B1 ). Then, since H 4 D,r (B 1 ) embeds compactly in C 1 D,r ( B1 ), we may regard the operator λA -1 g ′ (u) ∈ L(C 1 D,r ( B1 ), H 4 D,r (B 1 )) as a compact operator in C 1 D,r ( B1 ) for each (λ, u) ∈ R × O. Hence, by
F u (λ, u) = 1 + λA -1 g ′ (u) , (λ, u) ∈ R × O ,
is a Fredholm operator of index 0. The remark in [5, p. 246] now entails that (C3)-(C5) therein hold. Next, we introduce the function

ν : (0, ∞) × O → [0, ∞) , (λ, u) → 1 minB 1 {1 + u} .
To verify (C6) from [START_REF] Buffoni | The sub-harmonic bifurcation of Stokes waves[END_REF] consider a sequence

(λ n , u n ) n∈N in (0, ∞) × O with F (λ n , u n ) = 0 and ν(λ n , u n ) ≤ c < ∞ for each n ∈ N. Then, by Lemma 2.2, each u n belongs to C 4 D,r ( B1 ) with u n ≥ -1 + c -1 in B1 and satisfies B∆ 2 u n -T ∆u n = -λ n g(u n ) in B 1 .
The above uniform lower bound on u n and the finiteness of λ * established in Lemma 2.3 now imply that the sequence

(λ n g(u n )) n∈N is bounded in L ∞ (B 1 ). Thus, (λ n , u n ) n∈N is bounded in (0, λ * ] × H 4 D,r (B 1 ) and so (λ n , u n ) n∈N has a converging subsequence in [0, λ * ] × C 1 D,r (
B1 ). Hence (C6) in [START_REF] Buffoni | The sub-harmonic bifurcation of Stokes waves[END_REF] holds true. Setting λ := 0, we clearly have ( λ, 0) ∈ (0, ∞) × O and ( λ, 0) is in the closure of A 0 defined in (2.6), whence (C7) in [START_REF] Buffoni | The sub-harmonic bifurcation of Stokes waves[END_REF]. Finally, suppose that

(λ n , u n ) n∈N is a sequence in (0, ∞) × O with F (λ n , u n ) = 0 and ν(λ n , u n ) ≤ c < ∞ for each n ∈ N, which converges in R × C 1 D,r ( B1 ) towards (λ, u) ∈ (0, ∞) × O. Then -1 + c -1 ≤ u n ≤ 0 in B 1 and 0 ≤ λ n ≤ λ * for each n ∈ N by Lemma 2.3, which entails that -1 + c -1 ≤ u ≤ 0, whence (λ, u) ∈ [0, λ * ] × O and F (λ, u) = 0. Since (λ, u) ∈ (0, ∞) × O,
this is only possible if (λ, u) = (0, 0). The implicit function theorem guarantees that (λ n , u n ) ∈ A 0 for n large enough. This yields (C8) in [START_REF] Buffoni | The sub-harmonic bifurcation of Stokes waves[END_REF], and therefore, we are in a position to apply [5, Theorem 2.4] and obtain: Theorem 2.5. There is a continuous function (Λ, U) : (0, ∞) → (0, ∞) × C 1 D,r ( B1 ) with the following properties:

(i) U(s) ∈ S Λ(s) r for each s ∈ (0, ∞); (ii) (Λ, U)((0, 1)) ⊂ A 0 and lim s→0 (Λ(s), U(s)) = (0, 0); (iii) (Λ, U) is injective on (Λ, U) -1 (S) and lim s→∞ min B1 U(s) = -1 ; (iv) at all points s ∈ (Λ, U) -1 (S), (Λ, U) is real analytic with Λ ′ (s) = 0.
Actually, more precise information is given in [5, Theorem 2.4] about the curve

A := {(Λ(s), U(s)) : s ∈ (0, ∞)} , (2.7)
traced out by the function (Λ, U), in particular, that it is piecewise analytic:

Remark 2.6. The set (Λ, U) -1 ( S \ S) ⊂ (0, ∞) consists of isolated values and locally near each point s 0 ∈ (Λ, U) -1 ( S \ S), there is a re-parametrization ζ of the parameter s such that (Λ, U) • ζ is real analytic with derivative vanishing possibly only at 0.

Before analyzing further the curve A and in particular showing that it "globally" extends A 0 defined in (2.6) (note that at this point, the curves A 0 and A could still coincide), we first derive general properties of solutions to (1.4)-(1.5) in the next subsection .

2.2. Properties of stationary solutions. We first recall the following sign-preserving property of the operator B∆ 2 -T ∆ with homogeneous clamped boundary conditions in B 1 with radial symmetry established in [START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF].

Lemma 2.7. Consider f ∈ C r ( B1 ) and w ∈ C 4 r (B 1 )∩C 2 r ( B1 ) such that w is a classical solution to B∆ 2 w -T ∆w = f in B 1 , w = ∂ ν w = 0 on ∂B 1 . Then, if f ≤ 0 in B 1 , either w ≡ 0 or w < 0 in B 1 . (2.8) Furthermore, min B1 w = w(0) , (2.9)
and there is r 0 ∈ (0, 1) such that ∆w < 0 in B1 \ Br 0 and ∆w > 0 in B r 0 \ {0} . Proof. The first statement (2.8) of Lemma 2.7 readily follows from [START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF]Theorem 1.4]. Furthermore, the proof of that result reveals that (2.10) is true. We next deduce from (2.10) that ∂ r r d-1 ∂ r w(r) < 0 for r ∈ (r 0 , 1] and ∂ r r d-1 ∂ r w(r) > 0 for r ∈ (0, r 0 ). Since ∂ r w(0) = ∂ r w(1) = 0 due to the radial symmetry of w, its regularity, and its boundary conditions, we conclude that ∂ r w(r) ≥ 0 for r ∈ [0, 1]. Then w is a non-decreasing function in [0, 1] and attains its minimum at r = 0.

Lemma 2.8. Define the scalar product

•, • on H 2 D,r (B 1 ) by v, w := B 1 [B∆v(x)∆w(x) + T ∇v(x) • ∇w(x)] dx , v, w ∈ H 2 D,r (B 1 )
.

Let K := v ∈ H 2 D,r (B 1 ) : v ≥ 0 be the positive cone of H 2 D,r (B 1
) and define its polar cone by

K • := {w ∈ H 2 D,r (B 1 ) : v, w ≤ 0 for all v ∈ K} . Then, given v ∈ H 2 D,r (B 1 ), there is a unique couple (v 1 , v 2 ) ∈ K × K • such that v 1 , v 2 = 0 and v = v 1 + v 2 . In addition, v 2 ≤ 0 a.e. in B 1 .
Proof. The fact that any v ∈ H 2 D,r (B 1 ) can be written in a unique way as a sum [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]. The nonnegativity property of v 2 actually follows from the sign-preserving property stated in Lemma 2.7 and can be proved as [START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF]Proposition 4.5], where only the one-dimensional case was handled.

v = v 1 + v 2 with v 1 , v 2 = 0 and (v 1 , v 2 ) ∈ K × K • is a well-known result due to Moreau
As in [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF] the linear stability of stationary solutions is an important tool in the detailed analysis to follow. For u ∈ S λ r , it is measured by

µ 1 (u) := inf B 1 B|∆v| 2 + T |∇v| 2 + λg ′ (u)v 2 dx : v ∈ H 2 D,r (B 1 ) , v 2 = 1 , (2.11
) which turns out to be a simple eigenvalue of the linearization of (2.1) when non-negative as shown in the following lemma. Lemma 2.9. Consider λ ∈ [0, λ * ] and u ∈ S λ r such that µ 1 (u) ≥ 0. Then the following hold:

(i) µ 1 (u) is a simple eigenvalue of the operator A + λg ′ (u) ∈ L(H 4 D,r (B 1 ), L 2,r (B 1 )) with a positive eigenfunction in C 4 D,r ( B1 ); (ii) µ 1 (u) > 0 if and only if F u (λ, u) = 1 + λA -1 g ′ (u) ∈ L(C 1 D,r ( B1 )) is boundedly invertible.
Proof. (i) A classical compactness argument along with the weak lower semicontinuity of the scalar product •, • in H 2 D,r (B 1 ) defined in Lemma 2.8 guarantee the existence of a minimizer φ to (2.11)

in H 2 D,r (B 1 ) satisfying φ 2 = 1. Then φ ∈ H 4 D,r (B 1
) is a solution to the corresponding Euler-Lagrange equation

B∆ 2 φ -T ∆φ + (λg ′ (u) -µ 1 (u)) φ = 0 in B 1 , φ = ∂ ν φ = 0 on ∂B 1 .
(2.12)

Now, let φ ∈ H 4 D,r (B 1 ) be any solution to the boundary value problem (2.12). According to Lemma 2.8, there is a unique couple (φ 1 , φ 2 ) ∈ K ×K • such that φ = φ 1 +φ 2 , φ 1 , φ 2 = 0, and φ 2 ≤ 0 a.e. in B 1 . We deduce from the definition (2.11) of µ 1 (u), the orthogonality properties of (φ 1 , φ 2 ), and (2.12) that

µ 1 (u) φ 1 -φ 2 2 2 ≤ φ 1 -φ 2 , φ 1 -φ 2 + λ B 1 g ′ (u)(φ 1 -φ 2 ) 2 dx ≤ φ 1 + φ 2 , φ 1 + φ 2 + λ B 1 g ′ (u)(φ 1 -φ 2 ) 2 dx ≤ λ B 1 g ′ (u) (φ 1 -φ 2 ) 2 -(φ 1 + φ 2 ) 2 dx + µ 1 (u) φ 1 + φ 2 2 2 , whence 0 ≤ -4λ B 1 g ′ (u)φ 1 φ 2 dx + 4µ 1 (u) B 1 φ 1 φ 2 dx .
Both terms of the right-hand side of the above inequality being non-positive, we infer from the negativity of g ′ that φ 1 φ 2 = 0 a.e. in B 1 .

(2.13) Now, for i = 1, 2, it follows from the embedding of H 2 (B 1 ) in C α ( B1 ) for α ∈ (0, 1) (recall that d ∈ {1, 2}) that φ i ∈ C α r ( B1 ) and, according to [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]Theorem 2.19], the boundary value problem

B∆ 2 ψ i -T ∆ψ i = [µ 1 (u) -λg ′ (u)] φ i in B 1 , ψ i = ∂ ν ψ i = 0 on ∂B 1 , (2.14)
has a unique classical radially symmetric solution

ψ i ∈ C 4+α r ( B1 ). Since µ 1 (u) -λg ′ (u) > 0, φ 1 ≥ 0 and φ 2 ≤ 0 in B 1 , it follows from Lemma 2.7 that ψ 1 ≥ 0 ≥ ψ 2 in B 1 with ψ 1 > 0 in B 1 if φ 1 ≡ 0 and ψ 2 < 0 in B 1 if φ 2 ≡ 0.
In addition, due to (2.12) and (2.14),

B∆ 2 (φ -ψ 1 -ψ 2 ) -T ∆(φ -ψ 1 -ψ 2 ) = [µ 1 (u) -λg ′ (u)] (φ -φ 1 -φ 2 ) = 0 in B 1 with φ -ψ 1 -ψ 2 ∈ H 4 D,r (B 1 ), whence φ = ψ 1 + ψ 2 .
Furthermore, ψ 1 clearly belongs to K while, for any v ∈ K, we infer from (2.14) that

ψ 2 , v = B 1 (µ 1 (u) -λg ′ (u)) φ 2 v dx ≤ 0 , so that ψ 2 ∈ K • .
The uniqueness of Moreau's decomposition then warrants that ψ i = φ i for i = 1, 2. Therefore, if φ 1 ≡ 0 and φ 2 ≡ 0, we deduce from the above analysis that ψ 1 ψ 2 < 0 a.e. in B 1 and ψ 1 ψ 2 = φ 1 φ 2 = 0 a.e. in B 1 , and a contradiction. Therefore, either φ 1 ≡ 0 or φ 2 ≡ 0, and we have shown that φ does not change sign in B 1 .

Consequently, any element of the kernel of the operator A + λg ′ (u) -µ 1 (u) in H 4 D,r (B 1 ) does not change sign, which implies that the kernel's dimension is one by a classical argument. Indeed, assume for contradiction that there are two linearly independent positive functions φ and ψ in the kernel. Then φ -αψ with suitable α > 0 is a sign-changing function in the kernel, which is impossible. Therefore, the kernel of A + λg ′ (u) is spanned by a positive function φ ∈ C 4 D,r ( B1 ), the additional regularity stemming from Lemma 2.2. Finally, to show that µ 1 (u) is a simple eigenvalue of A + λg ′ (u), consider Φ ∈ H 4 D,r (B 1 ) such that AΦ ∈ H 4 D,r (B 1 ) and (A + λg ′ (u) -µ 1 (u)) 2 Φ = 0. Then, (A + λg ′ (u) -µ 1 (u))Φ = αφ for some α ∈ R. Multiplying this identity by φ and integrating over B 1 gives α φ 2 2 = 0, thus α = 0. This yields assertion (i).

(ii) Assume that F u (λ, u) = 1 + λA -1 g ′ (u) ∈ L(C 1 D,r ( B1 )) is not boundedly invertible. Then -1 is an eigenvalue of the compact operator λA -1 g ′ (u) ∈ L(C 1 D,r ( B1 )). Hence there is φ ∈ C 1 D,r ( B1 ) with φ + λA -1 g ′ (u)φ = 0. Alternatively, Aφ = -λg ′ (u)φ so that φ ∈ H 4 D,r (B 1 ) by Lemma 2.2 and µ 1 (u) φ 2 2 ≤ φ, φ + λ B 1 g ′ (u)φ 2 dx = 0 , which implies µ 1 (u) ≤ 0.
Conversely, if µ 1 (u) = 0, then, arguing as in the proof of Lemma 2.9, we obtain a solution φ ∈ H 4 D,r (B 1 ) to

B∆ 2 φ -T ∆φ + λg ′ (u)φ = 0 in B 1 ,
and thus φ + λA -1 g ′ (u)φ = 0.

As in [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF]Chapter 11], a key tool in the analysis is the following comparison lemma.

Lemma 2.10. Consider λ ∈ (0, λ * ] and u ∈ S λ r such that µ 1 (u) ≥ 0.

(i) If v ∈ C 4 D,r (B 1 ) ∩ C 2 r ( B1 ) is a classical subsolution to (2.1)-(2.2) with v > -1 in B1 , then v ≤ u in B 1 . (ii) Furthermore, v = u if µ 1 (u) = 0.
Proof. (i) We proceed along the lines of the proof of [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF]Lemma 11.3.4]. By Lemma 2.8 there is a unique couple (w

1 , w 2 ) ∈ K × K • such that v -u = w 1 + w 2 , w 1 , w 2 = 0, and w 2 ≤ 0 a.e. in B 1 . Since B∆ 2 (v -u) -T ∆(v -u) + λ(g(v) -g(u)) ≤ 0 in B 1
and w 1 ∈ K, we may multiply the above inequality by w 1 and integrate over B 1 to obtain

w 1 , v -u + λ B 1 (g(v) -g(u))w 1 dx ≤ 0 . We next deduce from µ 1 (u) ≥ 0 that w 1 , v -u = w 1 , w 1 ≥ -λ B 1 g ′ (u)w 2 1 dx = -λ B 1 g ′ (u)w 1 (v -u -w 2 ) dx = -λ B 1 g ′ (u)w 1 (v -u) dx + λ B 1 g ′ (u)w 1 w 2 dx .
Combining the previous two inequalities gives

λ B 1 (g(v) -g(u) -g ′ (u)(v -u))w 1 dx + λ B 1 g ′ (u)w 1 w 2 dx ≤ 0 .
Owing to the convexity and the monotonicity of g together with the sign properties of w 1 and w 2 , the two terms on the left-hand side of the above inequality are non-negative. Therefore,

(g(v) -g(u) -g ′ (u)(v -u))w 1 = w 1 w 2 = 0 a.e. in B 1
and, in particular,

g(v) -g(u) -g ′ (u)(v -u) = w 2 = 0 a.e. in {x ∈ B 1 : w 1 (x) > 0} .
Since g is strictly convex, this implies that v -u = w 2 = 0 a.e. in {x ∈ B 1 : w 1 (x) > 0}. We have thus shown that v -u = 0 a.e. in {x ∈ B 1 :

w 1 (x) > 0} and, since v -u = w 2 ≤ 0 a.e. in {x ∈ B 1 : w 1 (x) = 0}, we conclude that v -u ≤ 0 a.e. in B 1 .
(ii) As in [10, Lemma 11.3.4], we define

f (ϑ) := ϑv + (1 -ϑ)u, φ + λ B 1 g(ϑv + (1 -ϑ)u)φ dx , ϑ ∈ [0, 1] ,
where φ is the unique positive eigenfunction of the linearized operator B∆ 2 -T ∆ + λg ′ (u) in H 4 D,r (B 1 ) associated to the eigenvalue µ 1 (u) = 0 satisfying φ 1 = 1, see Lemma 2.9. Since g is convex, φ > 0 in B 1 , and ϑv + (1 -ϑ)u satisfies

B∆ 2 (ϑv + (1 -ϑ)u) -T ∆(ϑv + (1 -ϑ)u) + λ (ϑg(v) + (1 -ϑ)g(u)) ≤ 0 in B 1 , we conclude f (ϑ) ≤ 0 , ϑ ∈ [0, 1] . (2.15) As f ′ (ϑ) = v -u, φ + λ B 1 g ′ (ϑv + (1 -ϑ)u)(v -u)φ dx and f ′′ (ϑ) = λ B 1 g ′′ (ϑv + (1 -ϑ)u)(v -u) 2 φ dx ,
the assumption µ 1 (u) = 0 guarantees that f ′ (0) = 0 while the convexity of g and the positivity of φ imply that f ′′ (0) ≥ 0. In addition, recalling that f (0) = 0, we deduce from (2.15) that f ′′ (0) ≤ 0. Therefore, f ′′ (0) = 0 and the strict convexity of g and the positivity of

φ in B 1 entail v = u.
In order to study more precisely the behavior of solutions to S λ r as the parameter λ varies, we now derive several estimates.

Lemma 2.11. There is

C 1 > 0 such that u H 2 + u C 3/2 ( B1 ) + λ B 1 g(u(x)) dx ≤ C 1 (2.16)
whenever λ ∈ [0, λ * ] and u ∈ S λ r . Proof. According to Lemma 2.2 there are m 1 > 0 and

φ 1 ∈ C 4 r ( B1 ) satisfying Aφ 1 = m 1 φ 1 and φ 1 > 0 in B 1 , φ 1 1 = 1 .
(2.17) Multiplying (2.1) by φ 1 and integrating over B 1 give

-m 1 B 1 φ 1 u dx = λ B 1 g(u)φ 1 dx . Since u ≥ -1 in B 1 , we deduce from (2.17) that 0 ≤ λ B 1 g(u)φ 1 dx ≤ m 1 . (2.18)
Next, recall that Lemma 2.7 ensures that the function u : [0, 1] → R defined by u(|x|) = u(x) for x ∈ B1 is non-decreasing. This readily implies that g(u) is non-increasing and, thanks to (2.18),

0 ≤ λ B 1 g(u(x)) dx = λ|∂B 1 | 1 0 g(u(r))r d-1 dr ≤ λ|∂B 1 | 3/4 0 g(u(r))r d-1 dr + λ|∂B 1 | 1 3/4 g(u(r -1/4))r d-1 dr ≤ λ|∂B 1 | 3/4 0 g(u(r))r d-1 dr + 2 d-1 λ|∂B 1 | 3/4 1/2 g(u(r))r d-1 dr ≤ λ(1 + 2 d-1 )|∂B 1 | 3/4 0 g(u(r))r d-1 dr ≤ 2 d λ minB 3/4 φ 1 B 3/4 φ 1 (x)g(u(x)) dx ≤ 2 d m 1 minB 3/4 φ 1 .
We have thus proved that λ g(u

) 1 ≤ C 1 . (2.19)
It next follows from (2.1), (2.19), and the non-negativity of g and 1 + u that

c u H 2 ≤ u, u = -λ B 1 g(u)u dx ≤ λ g(u) 1 ≤ C 1 . (2.20)
Finally, if d = 1, the embedding of H 2 (B 1 ) in C 3/2 ( B1 ) completes the proof in this case. If d = 2, we note that u solves B∆ 2 u = T ∆u -λg(u) in B 1 subject to homogeneous Dirichlet boundary conditions with T ∆u -λg(u) 1 ≤ C by (2. [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]) and (2.20). Hence, in this case the assertion follows from a version of the Brezis-Merle inequality [START_REF] Brezis | Uniform estimates and blow-up behavior for solutions of -∆u = V (x)e u in two dimensions[END_REF] (see Lemma A.1) and the embedding of W 2 q (B 1 ) in C 3/2 ( B1 ) for q large enough.

Restricting our attention to u ∈ S λ r with a non-negative µ 1 (u), the previous estimates can be improved in the following way. Lemma 2.12. There is

C 2 > 0 such that u H 2 + B 1 dx (1 + u(x)) 3 ≤ C 2 (2.21)
whenever λ ∈ [0, λ * ] and u ∈ S λ r with µ 1 (u) ≥ 0. Proof. We infer from (2.1) and the assumption µ 1 (u) ≥ 0 that -λ

B 1 g(u)u dx = u, u ≥ -λ B 1 g ′ (u)u 2 dx and thus B 1 3u 2 + u (1 + u) 3 dx ≤ 0 . (2.22)
Observing that 3z 2 + z ≥ 1/4 for z ∈ (-1, -1/2), we deduce from (2.22) that

B 1 1 (1 + u) 3 dx ≤ B 1 1 (-1/2,∞) (u) (1 + u) 3 dx + B 1 1 (-1,-1/2) (u) (1 + u) 3 dx ≤ 8|B 1 | + 4 B 1 1 (-1,-1/2) (u)(3u 2 + u) (1 + u) 3 dx ≤ 8|B 1 | .
Finally, (2.1) and Hölder's inequality give

0 ≤ u, u = -λ B 1 g(u)u dx ≤ λ * |B 1 | 1/3 B 1 1 (1 + u) 3 dx 2/3
, and (2.21) follows from the previous two inequalities and the finiteness of λ * . Proposition 2.13. (i) For any λ ∈ (0, λ * ), the set S λ r is non-empty and contains a unique maximal element u λ in the sense that u ≤ u λ for all radially symmetric classical subsolutions u to (2.1)-(2.2) with parameter λ. In addition, for each x ∈ B 1 , the function λ → u λ (x) is decreasing in (0, λ * ).

(ii) There is no radially symmetric classical solution to (2.1)-(2.2) for λ > λ * .

We supplement Proposition 2.13 with continuity properties of λ -→ u λ .

Lemma 2.14. The map λ -→ u λ is continuous from [0, λ * ) to C 2 r ( B1 ) with u 0 = 0. In addition, the map λ -→ µ 1 (u λ ) belongs to C([0, λ * )).

Proof. Fix λ ∈ [0, λ * ) and let (λ k ) k≥1 be a sequence in [0, λ * ) such that λ k → λ as k → ∞. Then there is η ∈ (0, 1) such that λ < ηλ * and λ k ≤ ηλ * < λ * , k ≥ 1 . Proposition 2.13 ensures that u λ k ≥ u ηλ * for all k ≥ 1, so that (u λ k ) k≥1 ranges in a compact subset of (-1, 0]. Therefore, (g(u λ k )) k≥1 is bounded in L ∞ (B 1 ) and classical regularity results entail that (u λ k ) k≥1 is bounded in W 4 q (B 1 ) for all q ∈ (1, ∞), see [11, Theorem 2.20] for instance. The compactness of Sobolev's embedding then implies that a subsequence of (u λ k ) k≥1 (not relabeled) converges weakly in H 4 (B 1 ) and strongly in C 3 ( B1 ) to a function u ∈ H 4 D,r (B 1 ), which is a strong solution to (2.1)-(2.2) and satisfies u ≥ u ηλ * > -1 in B1 . Since g is smooth in (-1, ∞), there is α > 0 such that g(u) belongs to C 1+α ( B1 ) and a further use of classical elliptic regularity results guarantees that u actually belongs to S λ r , see [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]Theorem 2.19] for instance.

Consider now a radially symmetric classical subsolution

σ ∈ C 4 r (B 1 ) ∩ C 2 ( B1 ) to (2.1)-(2.
2) with σ > -1 in B1 . For ϑ ∈ (0, 1) and k ≥ 1, we infer from the convexity of g and the properties λ k ≤ ηλ * and g(u ηλ * ) ≥ 1 in B 1 that

B∆ 2 ((1 -ϑ)σ + ϑu ηλ * ) -T ∆ ((1 -ϑ)σ + ϑu ηλ * ) + λ k g ((1 -ϑ)σ + ϑu ηλ * ) ≤(1 -ϑ) B∆ 2 σ -T ∆σ + λ k g(σ) + ϑ B∆ 2 u ηλ * -T ∆u ηλ * + λ k g(u ηλ * ) ≤(1 -ϑ)(λ k -λ)g(σ) + ϑ(λ k -ηλ * )g(u ηλ * ) ≤|λ k -λ| g(σ) ∞ -ϑ(ηλ * -λ k ) . Since λ k → λ as k → ∞ and λ < ηλ * , there is k ϑ ≥ 1 large enough such that, for all k ≥ k ϑ , |λ k -λ| g(σ) ∞ -ϑ(ηλ * -λ k ) ≤ 0 ,
and hence (1 -ϑ)σ + ϑu ηλ * is a subsolution to (2.1)-(2.2) with parameter λ k . Therefore, owing to the maximality property of u λ k ,

(1 -ϑ)σ + ϑu ηλ * ≤ u λ k in B 1
for all k ≥ k ϑ . We first let k → ∞ and then ϑ → 0 in the above inequality to conclude that σ ≤ u in B 1 . In other words, u is a maximal solution to (2.1)-(2.2) and thus u = u λ .

Owing to the definition (2.11), the continuity of λ -→ µ 1 (u λ ) in [0, λ * ) readily follows from that of λ -→ u λ which we have just established.

The next proposition entails that the maximal solutions are exactly the linearly stable solutions.

Proposition 2.15. Let λ ∈ [0, λ * ). Then µ 1 (u λ ) > 0, and if u ∈ S λ r satisfies µ 1 (u) ≥ 0, then u = u λ .
Proof. Due to the monotonicity and negativity of g ′ and the monotonicity of λ -→ u λ stated in Proposition 2.13, it readily follows from (2.11) that

µ 1 (u λ 1 ) ≥ µ 1 (u λ 2 ) for 0 ≤ λ 1 ≤ λ 2 < λ * . Introducing λ st := sup {λ ∈ [0, λ * ) : µ 1 (u λ ) > 0} ,
we assume for contradiction that λ st < λ * . Lemma 2.14 then ensures that µ 1 (u λst ) = 0. Now, given λ ∈ (λ st , λ * ), we deduce from (2.1) that

B∆ 2 u λ -T ∆u λ + λ st g(u λ ) = (λ st -λ)g(u λ ) < 0 in B 1 .
Applying Lemma 2.10 (ii), we conclude that u λ = u λst and a contradiction. Therefore, λ st = λ * . Finally, considering u ∈ S λ r such that µ 1 (u) ≥ 0, Lemma 2.10 (i) implies u λ ≤ u while the maximal property of u λ guarantees u ≤ u λ . Therefore, u = u λ .

We now show that the maximal arc-connected set A 0 defined in (2.6) coincides with the branch of maximal solutions (λ, u λ ), λ ∈ (0, λ * ).

Corollary 2.16. There holds λ 0 = λ * and V (λ) = u λ for each λ ∈ (0, λ * ). Moreover, lim λ→λ * µ 1 (u λ ) = 0.

Proof. Since V (λ) is for each λ ∈ (0, λ 0 ) a classical radially symmetric solution to (2.1)-(2.2), we clearly have λ 0 ≤ λ * . We now claim that

µ 1 (V (λ)) > 0 for λ ∈ [0, λ 0 ) . (2.23) Indeed, the continuity of V entails that λ → µ 1 (V (λ)) is continuous on [0, λ 0 ) with µ 1 (V (0)) = µ 1 (0) = m 1 > 0 with m 1 introduced in Lemma 2.2.
Clearly, Lemma 2.9 (ii) implies µ 1 (V (λ)) > 0 for each λ ∈ [0, λ 0 ). Consequently, u λ = V (λ) for each λ ∈ [0, λ 0 ) by Proposition 2.15. Now, suppose for contradiction that λ 0 < λ * . Then V (λ) → u λ 0 in C 2 r ( B1 ) as λ → λ 0 according to Lemma 2.14. Owing to the maximality of λ 0 , this implies that F u (λ 0 , u λ 0 ) is not boundedly invertible which contradicts Proposition 2.15 since µ 1 (u λ 0 ) > 0. In particular, lim λ→λ * µ 1 (u λ ) = 0 due to the maximality λ * .

We finally investigate the behavior of u λ as λ → λ * . Proposition 2.17. For x ∈ B1 , define

u λ * (x) := inf λ∈(0,λ * ) u λ (x) = lim λ→λ * u λ (x) ∈ [-1, 0] .
(2.24)

Then u λ * ∈ S λ * r and µ 1 (u λ * ) = 0. Moreover, any u ∈ S λ * r satisfies u ≤ u λ * in B 1 and if, in addition, µ 1 (u) ≥ 0, then u = u λ * .
Proof. The proof is similar to that of [START_REF] Esposito | Mathematical analysis of partial differential equations modeling electrostatic MEMS[END_REF]Theorem 11.4.1]. Indeed, the fact that u λ * is welldefined is a simple consequence of Proposition 2.13 (i). Thanks to Proposition 2.15, we are in a position to apply Lemma 2.12 and conclude that (u λ ) λ is bounded in H 2 (B 1 ) while (g(u λ )) λ is bounded in L 3/2 (B 1 ). Consequently, the sequence (u λ ) λ is bounded in W 4 3/2 (B 1 ) by classical elliptic regularity, see [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]Theorem 2.20] for instance, so that u λ * ∈ C 1 ( B1 ) due to the continuous embedding of W 4 3/2 (B 1 ) in C 1 ( B1 ). If the minimum of u λ * in B 1 would be equal to -1, then u λ * (0) = -1 and ∇u λ * (0) = 0 according to Lemma 2.7. These properties entail that there is C > 0 such that

1 1 + u λ * (x) ≥ C |x| , x ∈ B1 ,
which contradicts the boundedness of (g(u λ )) λ in L 3/2 (B 1 ). Therefore, u λ * > -1 in B1 which, together with the above mentioned estimates and classical elliptic regularity, implies u λ * ∈ S λ * r . That µ 1 (u λ * ) = 0 follows from Corollary 2.16.

Finally, take u ∈ S λ * r . Then, for all λ ∈ (0, λ * ), Indeed, let us first note that (λ * , u λ * ) cannot be the limit of (Λ(s), U(s)) as s → ∞ owing to Proposition 2.17 and Theorem 2.5 (iii). Thus, there is a minimal

B∆ 2 u -T ∆u + λg(u) = (λ -λ * )g(u) ≤ 0 in B 1 ,
s * ∈ [1, ∞) such that (Λ(s * ), U(s * )) = (λ * , u λ * ) and A 0 = (Λ, U)((0, s * )) .
(2.25)

Next recall from Proposition 2.17 that u λ * ∈ S λ * r with µ 1 (u λ * ) = 0 and it thus readily follows from Lemma 2.9 that the kernel of the partial Fréchet derivative F u (λ * , u λ * ) of the function F defined in (2.5) is spanned by a positive function φ * ∈ C 4 D,r ( B1 ). Now, a precise description of the behavior of A near (λ * , u λ * ) can be obtained from [8, Theorem 3.2] and is stated in the following theorem.

Theorem 2.18. There are δ > 0, ε > 0, and an injective and continuous function ζ from (-δ, δ) onto (s * -ε, s * + ε) with the following properties:

(i) ζ(0) = s * ; (ii) (Λ, U) • ζ is a real analytic function on (-δ, δ); (iii) all solutions (λ, u) to (2.1)-(2.2) near the point (λ * , u λ * ) = (Λ(s * ), U(s * )) lie on the curve {(Λ, U) • ζ(σ) : |σ| < δ}; (iv) (Λ • ζ) ′ (0) = 0 and (U • ζ) ′ (0) = φ * .
Proof. Since u λ * ∈ S λ * r we have u λ * ∈ O with O defined in (2.4) and thus the function F is analytic near (λ * , u λ * ). As the kernel of F u (λ * , u λ * ) is one-dimensional, codim(rg(F u (λ * , u λ * ))) equals 1 since F u (λ * , u λ * ) is a Fredholm operator of index 0, see e.g. [START_REF] Rudin | Functional analysis[END_REF]Theorem 4.25]. We now claim that

F λ (λ * , u λ * ) = A -1 g(u λ * ) ∈ rg(F u (λ * , u λ * )) . Indeed, if otherwise there is ϕ ∈ C 1 D,r ( B1 ) with A -1 g(u λ * ) = ϕ + λ * A -1 g ′ (u λ * )ϕ . But then ϕ ∈ H 4 D,r (B 1 ) satisfies B∆ 2 ϕ -T ∆ϕ + λ * g ′ (u λ * )ϕ = g(u λ * ) in B 1 ,
and testing this equation with φ * > 0 yields the contradiction

0 = B 1 B∆ 2 φ * -T ∆φ * + λ * g ′ (u λ * )φ * ϕ dx = B 1 g(u λ * ) φ * dx > 0 .
Therefore, we are in a position to apply [8, Theorem 3.2] and obtain in combination with Theorem 2.5 the assertion.

Actually, the curve A bends down at (λ * , u λ * ):

Lemma 2.21. Let w be a radially symmetric function in H 2 D,r (B 1 ) such that g(w) ∈ L 1 (B 1 ), and let the profile w of w defined by w(|x|) := w(x) for x ∈ B1 be a non-decreasing function on [0, 1]. Then there is C 3 > 0 such that

I d (1 + w(0)) ≤ C 3 1 + w 2 H 2 g(w) 1 , (2.26)
where d = 1, 2 and

I 1 (z) := 1 z 4/3 -1 , I 2 (z) := -ln z , z ∈ (0, 1) .
Proof. Since w is non-decreasing on [0, 1], we deduce from the Cauchy-Schwarz inequality that, for r ∈ [0, 1],

0 ≤ r d-1 ∂ r w(r) = r 0 ∂ s s d-1 ∂ s w(s) ds ≤ r d d r 0 1 s d-1 ∂ s s d-1 ∂ s w(s) 2 s d-1 ds 1/2 ≤ r d/2 d|∂B 1 | ∆w 2 . Then 0 ≤ ∂ r w(r) ≤ C ∆w 2 r (2-d)/2
and integrating once more with respect to r gives

w(r) ≤ w(0) + C ∆w 2 r (4-d)/2 , r ∈ [0, 1] .
Consequently,

B 1 g(w(x))dx ≥ B 1 g w(0) + C ∆w 2 |x| (4-d)/2 dx .
Setting ̺ := (1 + w(0)) 2/(4-d) and restricting the integral on the right-hand side of the above inequality to B 1 \ B ̺ , we obtain

B 1 g(w(x))dx ≥ B 1 \B̺ 1 (1 + w(0) + C ∆w 2 |x| (4-d)/2 ) 2 dx ≥ B 1 \B̺ 1 (1 + C ∆w 2 ) 2 dx |x| (4-d) ≥ C (1 + ∆w 2 2 ) I d (1 + w(0)) , whence (2.26). Lemma 2.22. Let (λ n ) n≥1 be a sequence of real numbers in [0, λ * ] and (v n ) n≥1 be such that v n ∈ S λn r for each n ≥ 1. If lim n→∞ min B1 v n = -1 , (2.27)
then there are a subsequence of (λ n , v n ) n≥1 (not relabeled) and

ω ∈ C 4 r ( B1 \ {0}) ∩ C r ( B1 ) such that ω solves B∆ 2 ω(x) -T ∆ω(x) = 0 for x ∈ B 1 \ {0} , (2.28) ω(x) = ∂ ν ω(x) = 0 for x ∈ ∂B 1 , (2.29)
and

lim n→∞ λ n = 0 , (2.30)
lim n→∞ v n -ω C 1+α ( B1 ) == 0 , α ∈ [0, 1/2) , (2.31)
v n ⇀ ω in H 2 (B 1 ) , (2.32) ω(0) = -1 and ω(x) > -1 for x ∈ B 1 \ {0} .
(2.33)

Proof.
Step 1: Compactness. By Lemma 2.11 and the finiteness of

λ * , (v n ) n is bounded in H 2 (B 1 ) ∩ C 3/2 ( B1 ) and (λ n ) n is bounded in [0, λ * ].
The compactness of the embedding of

C 3/2 ( B1 ) in C 1+α ( B1 ) for α ∈ [0, 1/2)
guarantees that, after possibly extracting a subsequence, we may assume that there are

λ ∞ ∈ [0, λ * ] and ω ∈ H 2 (B 1 ) ∩ C 1+α ( B1 ) for α ∈ [0, 1/2) such that λ n → λ ∞ as n → ∞
(0) = -1 , ∇ω(0) = 0 , and ω = ∂ ν ω = 0 on ∂B 1 . (2.34)
In addition, by Lemma 2.7, v n is radially symmetric with a non-decreasing profile v n defined by v n (|x|) := v n (x) for x ∈ B1 . Consequently, the function ω enjoys the same properties by (2.31) and its profile v, defined by v(|x|) := ω(x) for x ∈ B1 , is a non-decreasing function on [0, 1]. Therefore, it follows from this property and (2.34) that there is a ∈ [0, 1) such that ω(x) = -1 for x ∈ Ba and ω(x) > -1 for x ∈ B 1 \ Ba .

(2.35)

In addition, if a > 0, then

∂ ν ω(x) = 0 for x ∈ ∂B a . (2.36)
Step 2: Identification of λ ∞ . To this end, we apply Lemma 2.21 and use Lemma 2.11 to obtain

λ n I d (1 + v n (0)) ≤ C 3 1 + v n 2 H 2 λ n g(v n ) 1 ≤ C 1 C 3 (1 + C 2 1 ) , (2.37)
which also reads

λ n 1 -(1 + v n (0)) 4/3 (1 + v n (0)) 4/3 ≤ C if d = 1 and λ n | ln (1 + v n (0))| ≤ C if d = 2 .
Letting n → ∞ in the above inequality readily gives λ ∞ = 0 by (2.27), whence (2.30). Next, fix ̺ ∈ (a, 1). We infer from (2.31) and (2.35) that (g (2.30). Classical elliptic regularity estimates then allow us to pass to the limit as n → ∞ and conclude that ω ∈ C 4 ( B1 \ B ̺ ) satisfies B∆ 2 ω(x) -T ∆ω(x) = 0 for all x ∈ B 1 \ B̺ and (2.29). Since ̺ is arbitrary in (a, 1), we have shown

(v n )) n≥1 is bounded in L ∞ (B 1 \ B̺ ), so that (λ n g(v n )) n≥1 converges to zero in L ∞ (B 1 \ B̺ ) as n → ∞ by
B∆ 2 ω(x) -T ∆ω(x) = 0 for x ∈ B 1 \ Ba .
(2.38)

Step 3: Identification of a. The final step is to prove that the yet unknown number a is equal to zero.

Let n ≥ 1. According to (2.10), there is r n ∈ (0, 1) such that ∆v n < 0 in B1 \ Brn and ∆v n > 0 in B rn .

(2.39)

The boundary conditions for v n and Lemma 2.11 then imply

B 1 \ Brn |∆v n (x)| dx = - B 1 \ Brn ∆v n (x) dx = - B 1 ∆v n (x) dx + Br n ∆v n (x) dx ≤ |B rn | ∆v n 2 ≤ C 1 |B rn | . ( 2
r n ≥ r ⋆ > 0 , n ≥ 1 . (2.41)
Now, for n ≥ 1 and x ∈ B1 , we set v n (|x|) = v n (x), w n (|x|) := ∆v n (x), and define

σ n := sup r ∈ (0, r ⋆ ) : v n (r) < -1 + λ n
if the set is non-empty, and σ n = 0 otherwise. Since σ n ∈ [0, r ⋆ ], we may assume, after possibly extracting a subsequence, that lim

n→∞ σ n = σ ∈ [0, r ⋆ ] . ( 2
B∆ 2 v n -T ∆v n = -λ n g(v n ) ≤ -1 in B σn . Consequently, ∂ r r d-1 (B∂ r w n (r) -T ∂ r v n (r)) ≤ -r d-1 , r ∈ (0, σ n ) ,
and, because r d-1 ∂ r w n (r) and r d-1 ∂ r v n (r) both vanish as r → 0, a first integration gives

∂ r (Bw n (r) -T v n (r)) ≤ - r d , r ∈ (0, σ n ) .
We next integrate the above differential inequality over (r, σ n ) to obtain

Bw n (r) ≥ Bw n (σ n ) + T (v n (r) -v n (σ n )) + σ 2 n 2d - r 2 2d .
Due to w n (σ n ) ≥ 0 by (2.43), we find

B∂ r r d-1 ∂ r v n (r) = Br d-1 w n (r) ≥ T r d-1 (v n (r) -v n (σ n )) + σ 2 n r d-1 2d - r d+1 2d ,
whence, after integrating once more,

Br d-1 ∂ r v n (r) ≥ T r 0 s d-1 (v n (s) -v n (σ n )) ds + σ 2 n r d 2d 2 - r d+2 2d(d + 2) , r ∈ (0, σ n ) .
Now, fix r ∈ (0, σ). Owing to (2.31) and (2.42), we may pass to the limit as n → ∞ in the above inequality and deduce

Br d-1 ∂ r v(r) ≥ T r 0 s d-1 (v(s) -v(σ)) ds + σ 2 r d 2d 2 - r d+2 2d(d + 2)
.

As r ∈ (0, σ), it follows from (2.43) that ∂ r v(r) = 0 and v(s) = -1 = v(σ) for s ∈ (0, r), so that we end up with

0 = Br d-1 ∂ r v(r) ≥ r d 2d 2 (d + 2) (d + 2)σ 2 -dr 2 > 0 ,
and thus a contradiction. We have thus shown that

lim n→∞ σ n = 0 . (2.44)
Let then r ∈ (0, 1) be arbitrary. Owing to (2.44), there is N r ≥ 1 large enough such that σ n ∈ (0, r) for n ≥ N r . Recalling the definition of σ n , this means that, for n ≥ N r and x ∈ B 1 \ Br , we have 1 + v n (x) ≥ √ λ n and thus

λ n g(v n ) ≤ 1 in B 1 \ Br . Since (v n ) n≥Nr is bounded in H 2 D (B
= 1 + v(a) = ∂ r v(a) = ∂ 2 r v(a) = ∂ 3 r v(a) .
(2.45) Multiplying (2.38) by ω, integrating over B 1 \ Ba and using (2.29) and (2.45) give

B 1 \ Ba B|∆ω| 2 + T |∇ω| 2 dx = 0 .
This implies that ω is constant in B 1 \ Ba and contradicts (2.29) and (2.45). Consequently, a = 0 and the proof is complete.

To finish off the proof of Theorem 2.20 it just remains to summarize our previous findings. Since Λ(s) → 0 for s → ∞ as just shown, we find for each λ ∈ (0, λ * ) numbers 0 < s 1 < s * < s 2 depending on λ such that Λ(s 1 ) = Λ(s 2 ) = λ. Since s 1 < s * , we have U(s 1 ) ∈ A 0 and thus U(s 1 ) ≥ U(s 2 ) by Proposition 2.13, Corollary 2.16, and (2.25). Moreover, U(s 1 ) = U(s 2 ) since no bifurcation can occur along the curve A 0 due to the implicit function theorem. This proves part (ii) of Theorem 2.20.

Theorem 1.1 and Remark 1.2 are now consequences of Theorem 2.5, Theorem 2.18, and Theorem 2.20.

WELL-POSEDNESS IN GENERAL DOMAINS

We shall now focus on the well-posedness of the dynamic problem. Let us recall that Ω is an arbitrary smooth domain in R d with d = 1, 2.

3.1.

Well-posedness for the hyperbolic problem. In this subsection, we prove Theorem 1.3. To lighten the notation, we agree upon setting γ = 1 in the following. We first reformulate (1.1)-(1.3) as a first-order Cauchy problem and use well-known results on cosine functions (see e.g. [1, Section 5.5 & Section 5.6] for details). Let us note that the self-adjoint operator -A = -B∆ 2 + T ∆ with domain H 4 D (Ω) generates an analytic semigroup on L 2 (Ω) with spectrum contained in [Re z < 0], its inverse A -1 is a compact linear operator on L 2 (Ω), and the square root of A is well-defined. Noticing that A is associated with the continuous coercive form

u, v = Ω (B∆u∆v + T ∇u • ∇v) dx , u, v ∈ H 2 D (Ω) ,
the domain of the square root of A is (up to equivalent norms) equal to H 2 D (Ω). Consequently, the matrix operator

A := 0 -1 A 1 with domain D(A) := H 4 D (Ω) × H 2 D (Ω)
generates a strongly continuous group on the Hilbert space

H := H 2 D (Ω) × L 2 (Ω). Writing u 0 = (u 0 , u 1 ), u = (u, ∂ t u), and
f (u) = 0 -g(u)
with g(u) := 1/(1 + u)

m = τ m (u 0 ) ∈ (0, ∞]. If τ m < ∞, then lim inf t→τm min Ω u(t) = -1 , (3.2
)

or lim sup t→τm (u(t), ∂ t u(t)) H = ∞ . (3.3)
To obtain more regularity on the mild solution u, let us consider an initial value in the domain of the generator -A, that is, let

u 0 ∈ H 4 D (Ω) × H 2 D (Ω) ∩ S(κ).
Then, since f is Lipschitz continuous, it follows as in the proof of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Theorem 6.1.6] that u : [0, τ m ) → H is Lipschitz continuous and whence differentiable almost everywhere with respect to time. Consequently, we obtain (see also [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]Corollary 4.2.11]):

Corollary 3.2. If u 0 ∈ H 4 D (Ω) × H 2 D (Ω) ∩ S(κ)
, then the mild solution u is actually a strong solution to (3.1). That is, u is differentiable almost everywhere in time with u ∈ L 1 (0, τ ; H) for each τ ∈ (0, τ m ) and u(t) = -Au(t) + f (u(t))

in H for almost every t ∈ [0, τ m ).

As a consequence, since u = (u, ∂ t u), we deduce under the assumption of Corollary 3.2 that, for each τ ∈ (0, τ m ),

∂ k t u ∈ C([0, τ m ), H 2-2k D (Ω)) , ∂ k+1 t u ∈ L 1 (0, τ ; H 2-2k D (Ω)) , for k = 0, 1 and (B∆ 2 -T ∆)u = -∂ 2 t u -∂ t u -λ(1 + u) -2 . ( 3.4)
Since the right-hand side of (3.4) belongs to L 1 (0, τ ; L 2 (Ω)), we deduce u ∈ L 1 (0, τ ;

H 4 D (Ω)). Now, testing (3.4) by ∂ t u ∈ C([0, τ m ), L 2 (Ω)) results in 1 2 d dt Ω |∂ t u| 2 dx + Ω B|∆u| 2 + T |∇u| 2 dx -2λ Ω 1 1 + u dx = - Ω |∂ t u| 2 dx (3.5)
almost everywhere in [0, τ m ). Assume now that τ m < ∞ and that (3.2) does not occur. Then (1 + u) -1 ∈ L ∞ ((0, τ m ) × Ω) so that (3.3) cannot occur as well, whence a contradiction. Consequently, τ m < ∞ implies (3.2).

To finish off the proof of Theorem 1.3, it remains to show that the solution exists globally in time for small λ and small initial values. Recall that H 2 (Ω) embeds continuously in L ∞ (Ω) since d = 1, 2 and let c 4 > 0 be such that

v 2 ∞ ≤ c 4 B ∆v 2 2 + T ∇v 2 2 , v ∈ H 2 D (Ω) .
Then we can prove the following result on global existence:

Corollary 3.3. For each κ ∈ (0, 1/2), there exists λ 1 (κ) > 0 such that τ m = ∞ provided that λ ≤ λ 1 (κ) and u 0 ∈ H 4 D (Ω) × H 2 D (Ω) ∩ S(2κ) with B ∆u 0 2 2 + T ∇u 0 2 2 + u 1 2 2 ≤ (1 -2κ) 2 c 4 .
Proof. Since u 0 ≥ -1 + 2κ, we have

T 0 := sup{τ ∈ (0, τ m ) : u(t) ≥ -1 + κ , t ∈ [0, τ )} > 0 and (1 + u(t)) -1 ≤ κ -1 for t ∈ [0, T 0 ). From (3.5), B ∆u(t) 2 2 + T ∇u(t) 2 2 ≤ B ∆u 0 2 2 + T ∇u 0 2 2 + u 1 2 2 + 2λ|Ω| κ (3.6)
for t ∈ [0, T 0 ) and therefore

u(t) 2 ∞ ≤ (1 -2κ) 2 + 2λc 4 |Ω| κ ≤ (1 -κ) 2 , t ∈ [0, T 0 ) , if λ ≤ λ 1 (κ) with λ 1 (κ) > 0 sufficiently small. Consequently, T 0 = τ m from which τ m = ∞ by Proposition 3.1.
Note that u ∈ L ∞ (0, ∞; H 2 D (Ω)) and u(t) ≥ -1 + κ for t ≥ 0 due to (3.6) and T 0 = ∞. Remark 3.4. If Ω = B 1 , the rotational invariance of (1.1) and the uniqueness of solutions guarantee that u(t) is radially symmetric for each t ∈ [0, τ m ) provided that (u 0 , u 1 ) is radially symmetric.

Remark 3.5. The proof of Theorem 1.3 is the same if the clamped boundary conditions (1.2) are replaced by the pinned boundary conditions (1.6), and we obtain a strong solution in this case as well. This improves the existence result for weak solutions in [START_REF] Guo | Dynamical solutions of singular wave equations modeling electrostatic MEMS[END_REF].

3.2.

Well-posedness for the parabolic problem. To prove Theorem 1.4 we first note that

g : {u ∈ H 2 D (Ω) : u ≥ -1 + κ in Ω} → H 2 (Ω) , u → (1 + u) -2 is uniformly Lipschitz continuous and recall that for instance H 2 (Ω) ֒→ H 1/4 D (Ω). We then also recall that the operator -A = -(B∆ 2 -T ∆) with domain H 4 D (Ω) generates an analytic semigroup {e -tA : t ≥ 0} on L 2 (Ω) with e -tA L(L 2 (Ω)) ≤ Me -αt , t ≥ 0 ,
for some α > 0. Formulating (1.7)-(1.9) by means of the variation-of-constant formula u(t) = e -tA u 0 -λ t 0 e -(t-s)A g(u(s)) ds , t ≥ 0 , the proof of Theorem 1.4 can be performed by a classical fixed point argument. In particular, the exponential decay of the semigroup entails global existence for small values of λ as stated in Theorem 1.4 (iii) (e.g. see [9, Theorem 1.2 (i)] for details). As in Remark 3.4, if Ω = B 1 , we easily see that u(t) is radially symmetric for each t ∈ [0, τ m ) provided that u 0 is radially symmetric.

TOUCHDOWN IN THE BALL

In this last section we return to the case Ω = B 1 and take advantage of the fact that a positive eigenfunction φ 1 > 0 to the operator A is available, see [START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF] and Lemma 2.2. We employ the eigenfunction method as e.g. in [START_REF] Chang | The quenching of solutions of semilinear hyperbolic equations[END_REF] to show the occurrence of a singularity in finite time as stated in Proposition 1.5 and Proposition 1.6. This completes the proof of Proposition 1.5.

4.2.

Radially symmetric initial data. Roughly speaking, the proof of Proposition 1.6 proceeds along the same lines of that of Proposition 1.5 but takes advantage of the properties of the linearization of (2.1) for λ * described in Lemma 2.9 and Proposition 2.17.

Proof of Proposition 1.6. Fix λ > λ * and let u be the maximal solution on [0, τ m ) to (1.1)-(1.3) if γ > 0 or (1.7)-(1.9) if γ = 0 corresponding to the initial value (u 0 , u 1 ). Recall that µ 1 (u λ * ) = 0 by Proposition 2.17 and that there exists a corresponding positive eigenfunction φ * ∈ C 4 D,r ( B1 ) to the operator A + λ * g ′ (u λ * ) according to Lemma 2.9, which we normalize so that φ * 1 = 1. For t ∈ [0, τ m ), define M(t) := Recalling that M(t) ≥ -1 for all t ∈ [0, τ m ) by (4.4), we end up with

-1 ≤ K 1,γ -(λ -λ * )g(K 0,γ )t , t ∈ [0, τ m ) .
Consequently,

τ m ≤ 1 + K 1,γ (λ -λ * )g(K 0,γ ) < ∞ ,
as claimed in Proposition 1.6. Now, let ϑ ∈ (0, 2/K 0 ). We argue as in the proof of [4, Theorem 1] and use (A.2), (A.3), the convexity of z → e ϑz , and Jensen's inequality to obtain Consider next q ∈ [1, ∞). We infer from (A.4) that

B 1 exp ϑ ∂ x i ∂ x j w(x) f 1 dx ≤ B 1 exp ϑ B 1 ∂ x i ∂ x j G(x, y) f (y) f 1 dy dx ≤ B 1 B 1 exp ϑ ∂ x i ∂ x j G(x, y) f (y) f 1 dydx ≤ B 1 f (y) f 1 B 1 2 + 1 |x -y|
∂ x i ∂ x j w q q ≤ f q 1 B 1 ∂ x i ∂ x j w(x) q f q 1 dx ≤ f q 1 sup z≥0 z q e -z/K 0 B 1 exp ∂ x i ∂ x j w(x) K 0 f 1 dx ≤C(1/K 0 ) f q 1 sup z≥0
z q e -z/K 0 , which, together with the Poincaré inequality, completes the proof.

Remark A.2. According to [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]Proposition 4.27], the estimate (A.2) is valid for an arbitrary smooth domain Ω of R 2 (with a constant depending on Ω) so that the validity of Lemma A.1 extends to arbitrary smooth domains of R 2 .

FIGURE 1 .

 1 FIGURE 1. Plot of the function ω for d = 2 and (B, T ) = (1, 50).

(2. 10 )

 10 Finally, the profile w of w defined by w(|x|) = w(x) for x ∈ B1 is a non-decreasing function on [0, 1].

2. 3 .

 3 Maximal stationary solutions. We first recall the existence of maximal solutions to (2.1)-(2.2) established in[START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF] Theorem 1.5].

Proof of Theorem 2 . 20 .

 220 Recall from Theorem 2.5, (2.25), Corollary 2.16, and (2.6) that A = {(Λ(s), U(s)) : s > 0} is a continuous curve with lim s→0 (Λ(s), U(s)) = (0, 0) , (Λ(s * ), U(s * )) = (λ * , u λ * ) , and A 0 = {(λ, u λ ) : λ ∈ (0, λ * )}. Part (i) of Theorem 2.20 now follows from Lemma 2.22 and Theorem 2.5 (iii).

2 ,Proposition 3 . 1 .

 231 we may reformulate (1.1)-(1.3) as a Cauchy problemu + Au = λf (u) , t > 0 , u(0) = u 0 (3.1) in H with u indicating the time derivative. Now, defining for κ ∈ (0, 1) the open subset S(κ) of H by S(κ) := {u ∈ H 2 D (Ω) : u > -1 + κ in Ω} × L 2 (Ω) , the function f : S(κ) → H is uniformly Lipschitz continuous. A classical argument then entails the following proposition. For each u 0 ∈ S(κ), the Cauchy problem (3.1) has a unique maximal mild solution u = (u, ∂ t u) ∈ C([0, τ m ), H) for some maximal time of existence τ

4. 1 . 1 φ 1 1 φ 1 2 ) 1 (B∆ 2 φ 1 - 1 φ 1

 1111121111 General initial conditions.Proof of Proposition 1.5. Recall that Lemma 2.2 ensures the existence of m 1 > 0 andφ 1 ∈ C 4 D,r ( B1 ) with φ 1 > 0 in B 1 , φ 1 1 = 1, and Aφ 1 = m 1 φ 1 . Let u be the maximal solution on [0, τ m ) to (1.1)-(1.3) if γ > 0 or (1.7)-(1.9) if γ = 0 corresponding to the initial value (u 0 , u 1 ) and define, for t ∈ [0, τ m ), N(t) := B (x)u(t, x) dx ≥ -B (x) dx = -1 .We multiply (1.1) by φ 1 , integrate over B 1 , and use the properties of φ 1 , the convexity of g, and Jensen's inequality to obtainγ 2 d 2 N dt 2 T ∆φ 1 )u dx -λg B u dx ≤ -m 1 N -λg(N) . (4.3) Setting χ(z) := m 1 z + λg(z) for z ∈ (-1, ∞), we note that χ is decreasing in (-1, z λ ) and increasing in (z λ , ∞) ,wherez λ := (2λ/m 1 ) 1/3 -1. The choice λ > 4m 1 /27 guarantees χ(z) ≥ χ(z λ ) > 0 for z ∈ (-1, ∞).We then infer from (4.1) and (4.3) that, for t ∈ [0, τ m ), dN dt (t) ≤ -χ(z λ ) and -1 < N(t) ≤ N(0) -χ(z λ )t , if γ = 0, respectively dN dt (t) ≤ e -t/γ 2 dN dt (0) + χ(z λ ) -χ(z λ ) and -1 < N(t) ≤ N(0) + γ 2 dN dt (0) + χ(z λ ) 1 -e -t/γ 2 -χ(z λ )t if γ > 0. Consequently, τ m ≤ 1 + N(0) + γ 2 dN dt (0) + χ(z λ ) 1 χ(z λ ) < ∞ .

B 1 u 1 φ 2 d 2 M dt 2 1 φ 1 φ 1 ( 1 φ 1 φ 1 g 1 φ

 112221111111 (t, x)φ * (x) dx ≥ -B * (x) dx = -1 .(4.4)As in[START_REF] Guo | Dynamical solutions of singular wave equations modeling electrostatic MEMS[END_REF] Theorem 4.1], we multiply (1.1) by φ * , integrate over B 1 , and use the equation satisfied by u λ * to obtainγ * B∆ 2 u -T ∆u + λg(u) dx + B * B∆ 2 u λ * -T ∆u λ * + λ * g(u λ * ) dx = -B u -u λ * ) B∆ 2 φ * -T ∆φ * dx -B * (λg(u) -λ * g(u λ * )) dx = B * [-λg(u) + λ * g(u λ * ) + λ * g ′ (u λ * )(u -u λ * )] dx .It follows from the convexity of g and Jensen's inequality thatλ * g(u λ * ) + λ * g ′ (u λ * )(u -u λ * ) ≤ λ * g(u)andB (u)φ * dx ≥ g(M) .Therefore, owing to the positivity of φ * and λ -λ * , * (-λg(u) + λ * g(u)) dx≤ -(λ -λ * )g(M) .(4.5)Since (λ -λ * )g(M) ≥ 0, a first consequence of (4.5) is that, for t ∈ [0, τ m ), dM dt (t) ≤ 0 and M(t) ≤ M(0)if γ = 0 or d dt e t/γ 2 dM dt (t) ≤ 0 and M(t) ≤ M(0) + γ 2 dM dt (0) 1 -e -t/γ 2 if γ > 0. In both cases, for t ∈ [0, τ m ), M(t) ≤ K 0,γ := M(0) + γ 2 dM dt (0) .(4.6)Recalling that g is decreasing, we deduce from (4.5) and (4.6) thatγ 2 d 2 M dt 2 + dM dt + (λ -λ * )g(K 0,γ ) ≤ 0 , whence, for t ∈ (0, τ m ), M(t) ≤ M(0) -(λ -λ * )g(K 0,γ )t if γ = 0 and M(t) ≤ M(0) -(λ -λ * )g(K 0,γ )t + γ 2 dM dt (0) + (λ -λ * )g(K 0,γ ) 1 -e -t/γ 2 if γ > 0.We have thus shown that, for t ∈ [0, τ m ), M(t) ≤ K 1,γ -(λ -λ * )g(K 0,γ )t , where K 1,γ := M(0) + γ 2 dM dt (0) + (λ -λ * )g(K 0,γ ) .

For 1 f

 1 (x, y) ∈ B1 × B1 , we have x ∈ B 2 (y) and thusB 1 exp ϑ ∂ x i ∂ x j w(x) f 1 dx ≤ B (y) f 1 B 2 (y) 5 |x -y| ϑK 0 dxdy ≤ C(ϑ) , (A.4)since -ϑK 0 > -2.

  and thus u ≤ u λ in B 1 by Proposition 2.13. Letting λ → λ * gives u ≤ u λ * . The uniqueness assertion is a consequence of Lemma 2.10 and the maximality of u λ * .2.4. Continuation of maximal solutions. We shall now completely characterize the radially symmetric solutions to (2.1)-(2.2) near (λ * , u λ * ). That is, we show that the curve A defined in (2.7) non-trivially extends the curve {(λ, u λ ) : λ ∈ (0, λ * )} of maximal solutions established in Subsection 2.3, the latter coinciding with the curve A 0 from (2.6) as proven in Corollary 2.[START_REF] Laurenc | Sign-preserving property for some fourth-order elliptic operators in one dimension and radial symmetry[END_REF]. Moreover, all radially symmetric solutions to (2.1)-(2.2) near (λ * , u λ * ) lie on this curve, which in particular excludes any bifurcation phenomenon near this point.

  and (2.31) and (2.32) hold true. Combining (2.27) and (2.31) readily gives ω

  .40) Assume for contradiction that there is a subsequence (r n k ) k of (r n ) n such that r n k → 0 as k → ∞. It readily follows from (2.40) that (∆v n k ) k converges to zero in L 1 (B 1 \ B̺ ) for all ̺ ∈ (0, 1). Recalling (2.32), we deduce that ∆ω = 0 almost everywhere in B 1 which, together with (2.29), implies ω ≡ 0 in B 1 and contradicts (2.35). Therefore, there is r ⋆ > 0 such that

  = -1 for x ∈ B σ and ∆v n (x) ≥ 0 for x ∈ B σn .

	.42)
	Assume σ > 0 for contradiction. The definition of σ n , (2.31), (2.39), and (2.41) then ensure
	that
	ω(x) (2.43) Since 1 + v n (x) ≤ √ λ n for x ∈ B σn and v n ∈ S λn r , we find

  1 ) by Lemma 2.11 and v n ∈ S λn r , we conclude that (v n ) n≥Nr is bounded in H 4 D (B 1 \ Br ). The convergence (2.31) then entails ω ∈ H 4 D (B 1 \ Br ) for all r ∈ (0, 1) . Assume now a > 0 for contradiction. Since ω ≡ -1 in B a , the just established regularity of ω leads us to 0

ACKNOWLEDGMENTS

This research was done while Ph.L. was enjoying the kind hospitality of the Institut für Angewandte Mathematik of the Leibniz Universität Hannover. The work of Ph.L. was partially supported by the Centre International de Mathématiques et d'Informatique CIMI, Toulouse.

Corollary 2. [START_REF] Moreau | Décomposition orthogonale d'un espace hilbertien selon deux cônes mutuellement polaires[END_REF]. There holds (Λ • ζ) ′′ (0) < 0.

Proof. Twice differentiation of the equality

with respect to σ at σ = 0 and Theorem 2.18 yield

. Testing this last equation with φ * > 0 and using the convexity of g imply

2.5. End point. The following theorem now completes the picture of the curve A defined in (2.7). It characterizes the limit of (Λ(s), U(s)) as s → ∞ and shows that for each λ ∈ (0, λ * ) there are at least two steady-states.

Theorem 2.20.

for each ρ ∈ (0, 1), where ω

(ii) For each λ ∈ (0, λ * ) there are at least two values 0 < s 1 < s * < s 2 with Λ(s j ) = λ, U(s j ) ∈ S λ r for j = 1, 2, and U(s 2 ) ≤ U(s 1 ) in B 1 with U(s 2 ) = U(s 1 ). Note that Theorem 2.20 allows one to compute the end point ω explicitly in terms of the modified Bessel functions of the first and second kinds for d = 2 and, respectively, in terms of the exponential function for d = 1 (cf. Figure 1).

To prove Theorem 2.20, we first need the following result relating the minimum of a function w to the integrability of g(w).

APPENDIX A. A BREZIS-MERLE ESTIMATE

We shall prove here in two space dimensions that solutions to the biharmonic equation with homogeneous Dirichlet boundary conditions and right hand sides in L 1 belong to W 2 q for any q ∈ (1, ∞) (a fact which is used in Lemma 2.11). This result is strongly reminiscent of the celebrated Brezis-Merle inequality [START_REF] Brezis | Uniform estimates and blow-up behavior for solutions of -∆u = V (x)e u in two dimensions[END_REF] stating that solutions to the Laplace equation with homogeneous Dirichlet boundary conditions and right hand sides in L 1 belong to L q for any q ∈ (1, ∞). The proof of Lemma A.1 below is actually very similar to that of [START_REF] Brezis | Uniform estimates and blow-up behavior for solutions of -∆u = V (x)e u in two dimensions[END_REF]Theorem 1] and is given merely for the sake of completeness. Let us point out that, because of the clamped boundary conditions (1.5), the result cannot be deduced directly from [4, Theorem 1] (in contrast to the case of pinned boundary conditions (1.6)).

) be the unique solution to

There are ϑ 0 > 0 and C 5 > 0 independent of f and w such that

where D 2 w is the Hessian matrix of w. Furthermore, given q ∈ [1, ∞), there is C 6 (q) > 0 independent of f and w such that w W 2 q ≤ C 6 (q) f 1 . Proof. A classical density argument allows us to assume that f ∈ C ∞ 0 (B 1 ). Introducing the Green function G associated with the operator ∆ 2 subject to homogeneous Dirichlet boundary conditions in B 1 , it follows from [START_REF] Gazzola | Polyharmonic boundary value problems[END_REF]Theorem 4.7] that there is K 0 > 0 such that The solution w to (A.1) can be written as

G(x, y)f (y) dy , x ∈ B1 , which further gives