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Some remarks on biharmonic elliptic problems with a singular nonlinearity ∗†‡ Baishun Lai 1 1 0 2 Abstract We study the following semilinear biharmonic equation n a J ∆2u= λ , in B, 1−u 0 u = ∂u = 0, on ∂B, 2 (cid:26) ∂n where B is the unit ball in Rn and n is the exterior unit normal vector. We ] P prove the existence of λ∗ > 0 such that for λ ∈ (0,λ∗) there exists a minimal A (classical) solutionu ,whichsatisfies0 < u < 1. Intheextremalcaseλ =λ∗, λ λ . h we prove the existence of a weak solution which is unique solution even in a t a veryweak sense. Besides,several newdifficultiesariseandmanyproblemsstill m remain to be solved. we list those of particular interest in the final section. [ 1 v 1 Introduction and results 4 0 9 In the last forty years a great deal has been written about existence and multiplicity 3 . of solutions to nonlinear second order elliptic problems in bounded and unbounded do- 1 0 mains of Rn(n ≥ 2). Important achievements on this topic have been made by applying 1 various combinations of analytical techniques, which include the variational and topolog- 1 : ical methods. For the latter, the fundamental tool which has been widely used is the v maximum principle. However, for higher order problems, a possible failure of the maxi- i X mum principle causes several technical difficulties, which attracted the interest of many r a researchers. In particular, recently fourth order equations with an singular non-linearity have been studied extensively. The motivation for considering these equations stems from a model for the steady states of a simple micro electromechanical system (MEMS) which has the general form (see for example [1]) α∆2u = (β |∇u|2dx+γ)∆u+ λf(x) in Ω, Ω (1−u)2(1+χR dx ) Ω (1−u)2 (M )  0 < u < 1 R in Ω, λ  u = α∂u = 0 on ∂Ω, ∂n where ∆2(·):= −∆(−∆) denotes the biharmonic operator, Ω ⊂ Rn is a smooth bounded domain, n denotes the outward pointing unit normal to ∂Ω and α,β,γ,χ ≥ 0, are phys- ically relevant constants, f ≥ 0 represents the permittivity profile, λ > 0 is a constant which is increasing with respect to the applied voltage. ∗Mathematics Subject Classification (2000). Primary: 35G30, 35J40. †Biharmonic operator, singular nonlinearity, minimal solutions, extremal solutions; ‡This work was supported by the the Natural Science Foundation of China (Grant No: 10971061). Take α = β = χ = 0 and γ = 1, one obtain a simple approximation of (M ) λ −∆u = λ f(x) in Ω, (1−u)2 0 < u < 1 in Ω, (Sλ)   u = 0 on ∂Ω. This simple model, which lends itself to the vast literature on second order semilinear eigenvalue problems, is already a rich source of interesting mathematical problems, see e.g. [2–4] and the references cited therein. The case where γ = β = χ = 0 and α = 1,f(x) ≡ 1 in the above model, that is when we replace (1−u)−2 with (1−u)−p ∆2u = λ in Ω, (1−u)p 0 < u < 1 in Ω, (P ) λ  u = ∂u = 0 on ∂Ω.  ∂n Becauseofthelackofa“maximumprinciple”, which playsuch acrucialroleindeveloping the theory for the Laplacian, for ∆2 with Dirichlet boundary condition in general domains (i.e., Ω 6= B), very little is known about (P ). As far as we are aware, only a paper [12] λ study this problem for general domains. However, if p > 1 and the Ω is a ball, (P ) λ has recently been studied extensively, see e.g. [5,7–10,13] and its references. One of the reasons tostudy (P )ina ball isthat a maximum principle holdsinthis situation, see [11], λ and so some tools that are well suited for (S ) can work for (P ). The second reason is λ λ 4 that one can easily find a explicit singular radial solution, denoted by 1−|x|p+1(p > 1), of (P ) for Ω = B and a suitable parameter λ which satisfy the first boundary condition λ but not the second. The singular radial solution, called “ghost” singular solution, play a fundamental role to characterize the “true” singular solution, see in particular [13]. In this paper, we will focus essentially our attention on the case where p = 1 and Ω is a ball, namely ∆2u = λ in B, 1−u 0 < u ≤ 1 in B, (1.1) λ  u = ∂u = 0 on ∂B.  ∂n For the corresponding second order problem, which is related to the general study of  singularities of minimal hypersurfaces of Euclidean space, has been studied by Meadows, see [14]. In that case, however, the start point was an explicit singular solution (i.e., u (r) = r2) with parameter λ = n−1. When turning to the biharmonic problem (1.1) , s λ one can not find any explicit singular solution even “ghost” singular solution which causes several technical difficulties. The first purpose of the present paper is to extend (1.1) λ some well-known results relative to (P ). The second (and perhaps most important) λ purpose of the present paper is to emphasize some striking differences between (1.1) and λ (P ). λ 1.1 Preliminaries Besides classical solution i.e. u ∈ C4(B¯) which satisfy (1.1) , let us introduce the λ class of weak solutions we will be dealing with. We denote by H2(B) the usual Sobolev 0 space which can be defined by completion as follows: H2(B) := cl{u ∈ C∞(B) : k∆uk < ∞} 0 0 2 and which is an Hilbert space endowed with the scalar product (u,v) := ∆u∆vdx H2(B) 0 B Z Definition 1.1 We say that u ∈ L1(B) is a weak solution of (1.1) provided 0 ≤ u ≤ 1 λ almost everywhere, 1 ∈ L1(B) and 1−u ϕ u∆2ϕdx = λ dx, ∀ϕ ∈ C4(B¯)∩H2(B) (1.1) (1−u) 0 B B Z Z When in (1.1) the equality is replaced by the inequality ≥ (resp.≤) and ϕ ≥ 0, we say that u is a weak super-solution (resp. weak sub-solution) of (1.1) provided the following boundary conditions are satisfied: u = 0 (resp.=) and ∂u ≤ 0 (resp.≥) on ∂B. ∂n Definition 1.2 We call a solution u of (1.1) minimal if u ≤ v a.e. in B for any further λ solution v of (1.1) λ If u is a classical solution of (1.1) , then it turns out to be well defined the linearized λ operator at u λ L := ∆2 − u (1−u)2 which yields the following notion of stability Definition 1.3 A classical solution u of (1.1) is semi-stable provided λ λϕ2 µ (u) = inf (∆ϕ)2 − : φ ∈ H2(B),kφk = 1 ≥ 0 1 (1−u)2 0 L2 B (cid:26)Z (cid:27) If µ (u) > 0 we say that u is stable. 1 As far as we are concerned with weak solutions, the linearized operator is no longer well defined, however we introduce the following weaker notion of stability. Definition 1.4 A weak solution u to (1.1) is said to be weakly stable if 1 ∈ L1(B) λ (1−u)2 and the following holds: λϕ2 |∆ϕ|2dx ≥ ,ϕ ∈ H2(B),ϕ ≥ 0. (1−u)2 0 B B Z Z According to the class of solutions which we consider, let us introduce the following values: λ∗ := sup{λ ≥ 0 : (1.1) posses a weak solution}; λ (1.2) λ := sup{λ ≥ 0 : (1.1) posses a classical solution}. ∗ λ Remark 1.1 Clearly, a classical solution is also a weak solution, so that one has λ ≤ λ∗. ∗ Moreover, by standard elliptic regularity theory for the biharmonic operator [15], any weak solution of (1.1) which satisfies ku k < 1 turns out to be smooth. λ λ Besides, we give a notion of H2(B)- weak solutions, which is an intermediate class 0 between classical and weak solutions. Definition 1.5 We say that u is a H2(B)- weak solution of (1.1) if (1 − u)−1 ∈ L1(B) 0 and if ∆u∆φ = λ φ(1−u)−1, ∀φ ∈ C4(B¯)∩H2(B). 0 B B Z Z We say that u is a H2(B)- weak super-solution (resp. H2(B)- weak sub-solution) of (1.1) 0 0 λ if for φ ≥ 0 the equality is replaced with ≥ (resp.≤) and u ≥ 0 (resp. ≤), ∂u ≤ 0 (resp. ∂n ≥) on ∂B. 1.2 Main results In order to state our results, we denote by ν the first eigenvalue of the biharmonic 1 operator on B with Dirichlet boundary conditions, which is characterized variationally as follows: ν := inf |∆u|2dx : u ∈ H2(B),kuk = 1 . 1 0 B (cid:26)Z (cid:27) It is well known that ν > 0, that it is simple, isolated and that the corresponding 1 eigenfunctions ψ > 0, spherically symmetric , radially decreasing and do not change sign. We may now state the following theorem. Theorem 1.1 There exists λ > 0 such that for 0 < λ < λ , (1.1) poses a minimal ∗ ∗ λ classical solution, denoted by u , which is positive and stable. Moreover, λ satisfies the λ ∗ following bounds: ν 1 max{4n(n−2),2n(n+2)} ≤ λ ≤ . ∗ 4 It is remarkable that at λ there is an immediate switch from existence of regular ∗ minimal solutions to nonexistence of any (even singular) solution. The only possibly singular minimal solution corresponds to λ = λ . This result is known from [16] for the ∗ second order problem(S ), but the method used there may not be carried over to fourth λ order problems. Nevertheless, the result extends to biharmonic case in the following theorem. Theorem 1.2 The following holds: λ = λ∗. ∗ In particular, for λ > λ∗ there are no solutions, even in the weak sense. Furthermore, for almost every x ∈ B, there exists u∗(x) := lim u (x) λ λ→λ∗ and u∗(x) is a weakly stable H2(B)− weak solution of (1.1) , which is called the extremal 0 λ solution. If n ≤ 4 then the extremal solution u∗ of (1.1)λ is smooth, i.e., u∗ = limλ→λ∗uλ(x) exists in the topology of C4(B). It is the unique regular solutions to (1.1)λ∗. From the above theorem, we note that the function u∗ exists in any dimension, dose solve (1.1)λ∗ intheH02(B)weak sense anditisaclassical solutionindimensions 1 ≤ n ≤ 4. This will allow us to start another branch of nonminimal (unstable ) solutions. Besides, inspired by [5,17,18] we get the following uniqueness of the extremal solution of (1.1) , λ which gives Theorem1.3. Theorem 1.3 Let v be a weak super-solution of (1.1) with parameter λ∗. Then v = u∗; λ in particular (1.1) has a unique weak solution. λ Fromthistheorem, weknowthattherearenostrictsuper-solutionstoequation(1.1)λ∗. Corollary 1.1 Let u ∈ H2(B) be a weak solution of (1.1) such that ku k = 1. Then λ 0 λ λ u is weakly stable if and only if λ = λ∗ and u = u∗ λ λ We may also characterize the uniform convergence to 0 of u as λ → 0 by giving the λ precise rate of its extinction. Theorem 1.4 For all λ ∈ (0,λ∗) let u be the minimal solution of (1.1) and let λ λ λ V (x) = 1−|x|2 2. λ 8n(n+2) (cid:2) (cid:3) Then u > V (x) for all λ < λ∗ and all |x| < 1, and λ λ u lim λ = 1 uniformly with respect to x ∈ B. λ→0 Vλ(x) 1.3 Key-ingredients Now we give some comparison principles which will be used throughout the paper Lemma 1.1 (Boggio’s principle, [11]) If u ∈ C4(B¯ ) satisfies R ∆2u ≥ 0 in B , R u = ∂u = 0 on ∂B , (cid:26) ∂n R then u ≥ 0 in B . R Lemma 1.2 Let u ∈ L1(B ) and suppose that R u∆2ϕ ≥ 0 B Z R for all ϕ ∈ C4(B¯R) such that ϕ ≥ 0 in BR, ϕ|∂BR = ∂∂ϕn|∂BR = 0. Then u ≥ 0 in BR. Moreover u ≡ 0 or u > 0 a.e., in B . R For a proof see Lemma 17 in [19]. From this lemma, we know that any solution of (1.1) is necessarily positive a.e. inside the ball. λ Lemma 1.3 If u ∈ H2(B ) is radial, ∆2u ≥ 0 in B in the weak sense, that is R R ∆u∆ϕ ≥ 0 ∀ϕ ∈ C∞(B ), ϕ ≥ 0 0 R B Z R and u|∂BR ≥ 0, ∂∂nu|∂BR ≤ 0 then u ≥ 0 in BR. Proof. For the sake of completeness, we include a brief proof here. We only deal with the case R = 1 for simplicity. Solve ∆2u = ∆2u in B 1 u = ∂u1 = 0 on ∂B (cid:26) 1 ∂n in the sense u ∈ H2(B) and ∆u ∆ϕ = ∆u∆ϕ for all ϕ ∈ C∞(B). Then u ≥ 0 in 1 0 B 1 B 0 1 B by Lemma 2.2. R R Let u = u−u so that ∆2u = 0 in B. Define f = ∆u . Then ∆f = 0 in B and since 2 1 2 2 f is radial we find that f is a constant. It follows that u = ar2+b. Using the boundary 2 conditions we deduce a+b ≥ 0 and a ≤ 0, which imply u ≥ 0. 2 Lemma 1.4 Let f ∈ L1(B ),f ≥ 0 almost everywhere. Then there exists a unique R u ∈ L1(B ) such that u ≥ 0 and R u∆2ϕ = fϕ, ϕ ∈ C4(B¯ )∩H2(B ). (1.3) R 0 R B B Z R Z R Moreover, there exists C > 0 which does not depend on f such that kuk ≤ Ckfk . 1 1 Proof. The proof is standard, see [19], we give a proof here for the sake of completeness. The uniqueness is clear. Indeed, let v and v be two solutions of (1.3). Then 1 2 v = v −v satisfies 1 2 v∆ϕ = 0 ϕ ∈ C4(B¯ )∩H2(B ). R 0 R B Z Given any ζ ∈ C∞(B) let ϕ be the solution of 0 ∆2ϕ = ζ in B, ϕ = ∂ϕ = 0 on ∂B. (cid:26) ∂n It follows that vζ = 0. B Z Since ζ is arbitrary, we deduce that v = 0. For the existence, Given an integer k ≥ 0 we set f = min{f(x),k}, so that f → f k k as k → ∞ in L1(B). Let v be the solution of k ∆2v = f in B, k k (1.4) v = ∂vk = 0 on ∂B. (cid:26) k ∂n The sequence (v ) is clearly monotone nondecreasing. It is also a cauchy sequence in k k≥0 L1(B) since (v −v ) = (f −f )ζ , k l k l 0 B B Z Z where ζ is defined by 0 ∆2ζ = 1 in B, 0 ζ = ∂ζ0 = 0 on ∂B. (cid:26) 0 ∂n Hence |v −v | ≤ C |f −f |dx. k l k l B B Z Z Passingtothelimitin(1.4)(aftermultiplicationbyϕ)weobtain(1.3)andu ≥ 0according to the Lemma 1.2. Finally, taking ϕ = ζ in (1.3), we obtain 0 kvk = v = fζ ≤ Ckfk L1 0 L1 B B Z Z and the proof is completed. Proposition 1.1 Assume the existence of a weak super-solution U of (1.1) . Then there λ exists a weak solution u of (1.1) so that 0 ≤ u ≤ U a.e in B. λ Proof. By means of a standard monotone iteration argument, set u := U and define 0 recursively u ∈ L1(B) as the unique solution of n+1 ϕ u ∆2ϕdx = λ dx,ϕ ∈ C4(B¯)∩H2(B), n+1 (1−u )2 0 B B n Z Z then we have (u −u )∆2ϕdx ≥ 0,ϕ ∈ C4(B¯)∩H2(B) n n+1 0 B Z and Lemma 2.2 yields 0 ≤ u ≤ u < U(x) a.e. for all n ∈ N. Since n+1 n (1−u )−1 ≤ (1−U)−1 ∈ L1(B), n and the claim follows from the Lebesgue convergence Theorem. We complete these preliminary results by proving a key lemma which provides a comparison principle. Lemma 1.5 Assume u is a weakly stable H2(B)- weak sub-solution of (1.1) and u is 1 0 λ 2 H2(B)- weak super-solution of (1.1) . Then, 0 λ (1) u ≤ u almost everywhere in B. 1 2 (2) if u is a classical solution such that µ (u) = 0 and U is any classical super-solution 1 of (1.1) , then u ≡ U. λ Proof. (1) Define ω := u − u . Then by the Moreau decomposition [10] for the 1 2 biharmonic operator, there exists ω ,ω ∈ H2(B), with ω = ω +ω ,ω ≥ 0 a.e., ∆2ω ≤ 0 1 2 0 1 2 1 2 in the H2(B)− weak sense and 0 ∆ω ∆ω = 0 1 2 B Z By Lemma 1.1, we have that ω ≤ 0 a.e. in B. 2 Given now 0 ≤ ϕ ∈ C∞(B), we have that 0 ∆ω∆ϕ ≤ λ (f(u )−f(u ))ϕ, 1 2 B B Z Z where f(u) = (1−u)−1. Since u is stable, one has λ f′(u)ω2 ≤ λ (∆ω )2 = λ ∆ω∆ω ≤ λ (f(u )−f(u ))ω 1 1 1 1 2 1 B B B B Z Z Z Z Since ω ≥ ω one also has 1 f′(u)ωω ≤ (f(u )−f(u ))ω 1 1 2 1 B B Z Z which once re-arrange gives ˜ fω ≥ 0, 1 B Z where f˜(u ) = f(u ) − f(u ) − f′(u )(u − u ). The strict convexity of f gives f˜ ≤ 0 1 1 2 1 1 2 and f˜< 0 whenever u 6= U. Since ω ≥ 0 a.e. in B one sees that ω ≤ 0 a.e. in B. The 1 inequality u ≤ u a.e. in B is then established. 1 2 (2) Let ϕ > 0 bethe first eigenfunction of ∆2−λf′(u) inH2(B), we now, for 0 ≤ t ≤ 1, 0 define g(t) = ∆(tU +(1−t)u)∆φ−λ f(tU +(1−t)u)φ, B B Z Z where φ is the above first eigenfunction. Since f is convex one sees that g(t) ≥ λ [tf(U)+(1−t)f(u)−f(tU +(1−t)u)]φ ≥ 0 B Z for every t ≥ 0. Since g(0) = 0 and g′(0) = ∆(U −u)∆φ−λf′(u)(U −u)φ = 0, B Z we get that g′′(0) = −λ f′′(u)(U −u)2φ ≥ 0. B Z Since f′′(u)φ > 0 in B, we finally get that U = u a.e. in B. (cid:3) 2 Existence results: proofs of Theorem 1.1 and 1.2 2.1 The branch of minimal solutions Let us define Λ := {λ ≥ 0 : (1.1) has a classical solution with parameter λ}. λ Proposition 2.1 For all 0 ≤ λ < λ , there exists a minimal classical solution u of ∗ λ (1.1) which is smooth and stable. Moreover, λ (i) The map λ → u , for λ ∈ (0,λ ) is differentiable and strictly increasing; λ ∗ (ii) The map λ → µ (u ) is decreasing on (0,λ ); 1 λ ∗ (iii) Let u˜ be a regular solution of (1.1) for λ ∈ (0,λ ), if u˜ is not the minimal λ λ ∗ λ solution, then µ (u˜ ) < 0. 1 λ Proof. First we show that Λ dose not consist of just λ = 0. To this end, let ψ be the R first eigenfunction of the biharmonic operator subject to Dirichlet boundary conditions on B ⊃ B which we normalize by sup Ψ = 1 and let ν > 0 be the corresponding R B R R R eigenvalue. Next, we are going to prove that for θ ∈ (0,1) the function ψ = θψ is a R super-solution of (1.1) as long as λ is sufficiently small. We have λ 0 < 1−θψ < 1, in B R and moreover λ λ ∆2ψ = ν θψ ≥ = R R 1−θψ 1−ψ R provide that ν θψ (1−θψ ) ≥ λ. R R R Notice that 0 < s := inf ψ < s := supψ < 1 1 2 x∈B x∈B and that ∂ψ < 0 on ∂B. Thus, looking at the function g(s) = s(1−s), for s ∈ [s ,s ], it ∂n 1 2 is easily seen that we can choose λ > 0 sufficiently small such that ν inf g(θψ(x)) > λ. R x∈B Since u ≡ 0 is a sub-solution of (1.1) , the classical sub-super solution Theorem provides λ a classical solution u to (1.1) . With such function u , we can use the Boggio principle λ λ λ to show straightforwardly that the iterative scheme ∆2u = λ in B, n,λ (1−un,λ)   un,λ = ∂u∂nn,λ = 0 in ∂B, (2.1)   u = 0 in B, 0,λ     gives rise to a monotone sequence {u } satisfying n,λ 0 = u ≤ u ...u ≤ ... ≤ u < 1 0,λ 1,λ n−1,λ λ for all n ∈ N. Therefore the minimal solution u is obtained as the increasing limit λ u (x) := lim u λ n,λ n→∞ Again from the Boggio positivity preserving property (Lemma 1.1) we obtain 0 < u < 1; λ in particular, from standard elliptic regularity theory for the biharmonic operator follows that u (x) is smooth. In order to prove stability, let us argue as follows: set λ λ := sup{λ ∈ (0,λ ) : µ (u ) > 0} ∗∗ ∗ 1 λ clearly λ ≤ λ . Now suppose by contradiction that λ < λ and let ε > 0 sufficiently ∗∗ ∗ ∗∗ ∗ small such that λ +ε < λ and v be the corresponding minimal solution. By the ∗∗ ∗ λ∗∗+ε definition and left continuity of the map λ → µ (u ) we have necessarily µ (u ) = 0. 1 λ 1 λ∗∗ Since v is a super-solution of (1.1) , by Lemma 1.5 we get v = u and thus λ∗∗+ε λ∗∗ λ∗∗+ε λ∗∗ ε = 0, a contradiction. Since each u is stable, then by setting F(u ,λ) := −∆2− λ , we get that F (u ,λ) λ λ 1−uλ uλ λ is invertible for 0 < λ < λ . It then follows from Implicit Function Theorem that u (x) ∗ λ is differentiable with respect to λ. Now we prove the map λ → u is strictly increasing on (0,λ ). Consider λ < λ < λ , λ ∗ 1 2 ∗ their corresponding minimal positive solutions u and u , and let u∗ be a solution for λ1 λ2 (1.1) . The same as the above iterative scheme, we have λ2 u = lim u (λ ;x) ≤ u∗ in B, λ1 n→∞ n 1 and in particular u ≤ u in B. Therefore, duλ ≥ 0 for all x ∈ B. λ1 λ2 dλ Finally, by differentiating (1.1) with respect to λ, and since λ → u is nondecreasing, λ λ we get du λ du λ du −∆2 λ − λ = ≥ 0, x ∈ B; λ = 0, x ∈ ∂B. dλ (1−u )2 dλ 1−u dλ λ λ Applying the strong maximum principle, we conclude that duλ > 0 on B for all 0 < λ < λ dλ ∗ That λ → µ is decreasing follow easily from the variational characterization of µ , 1,λ 1,λ the monotonicity of λ → u , as well as the monotonicity of (1−u )−2 with respect to u λ λ λ and the proof of the (ii) is completed. Now we give the proof of (iii). Let u be the minimal solution for (1.1) so that λ λ u˜ ≥ u . If the linearization around u˜ had nonnegative first eigenvalue, then Lemma 1.5 λ λ λ would also yield u˜ ≤ u so that u˜ and u necessarily coincide, a contradiction. λ λ λ λ Remark 2.1 Dose (iii) of Proposition (2.1) extend to weak solutions u as formulated in [21, Theorem 3.1]? 2.2 Weak solutions versus classical solutions Lemma 2.1 Let u be a weak solution of (1.1) with µ < λ∗. Then, for ε > 0 sufficiently µ µ small, the problem (1.1) posses a classical solution. (1−ε)µ Proof. Let u˜ ∈ L1(B) be the unique solution of (1−ε) u˜∆2ϕ = µ ϕdx, ϕ ∈ C4(B¯)∩H2(B) 1−u 0 B B µ Z Z provided by Lemma 1.4. By hypothesis we have ε u ∆2ϕdx = µ ϕdx, ϕ ∈ C4(B¯)∩H2(B). µ 1−u 0 B B µ Z Z By uniqueness we get (1−ε)u = u˜ µ whereas Lemma 1.2 yields u˜ > 0 a.e. in B and hence we may assume u > u˜, x ∈ B\{x ∈ B : u˜ = 0} µ Therefore, (1−ε)µ 1 u˜∆2ϕ = dx ≥ (1−ε)µ dx, ϕ ∈ C4(B¯)∩H2(B) 1− 1 u˜ 1−u˜ 0 B B B Z Z 1−ε Z thus u˜ is a weak super-(cid:0)solution o(cid:1)f (1.1) and Proposition 2.1 yields a weak solution v (1−ε)µ of (1.1) which satisfies (1−ε)µ 0 ≤ v ≤ u˜ < u ≤ 1 µ and then classical by Remark 1.1. Remark 2.2 From this Lemma, we know that λ∗ = λ , in what follows, we always denote ∗ by λ the largest possible value of λ such that (1.1) has a solution, unless otherwise stated. ∗ λ Proposition 2.2 Up to a subsequence, the convergence u∗ := lim u (x) λ λրλ∗ holds in H02(B) and the extremal solution uλ∗ satisfies 1 ∆u∗∆ϕ = λ , ϕ ∈ C∞(B). (2.2) ∗ (1−u∗) 0 B B Z Z In particular, the extremal solution is weakly stable and if ku∗k < 1 then µ (u∗) = 0. ∞ 1