SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM.

Transcription

1 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. MAYTE PÉREZ-LLANOS AND ANA PRIMO Abstract. The aim of this work is to study the optimal exponent p to have solvability of problem 2 u = λ u + u p + cf in, u > 0 in, u = u = 0 on, where p > 1, λ > 0, c > 0, and R N, N >, is a smooth and bounded domain such that 0. We consider f 0 under some hypothesis that we will precise later. 1. Introduction The main concern of this work is to determine a critical threshold exponent p to have solvability of the following problem, 2 u = λ u + u p + cf in, (1) u > 0 in, u = u = 0 on, where p > 1, λ > 0, c > 0, and R N, N >, is a smooth and bounded domain such that 0. There exists a large literature dealing with the case λ = 0. The differential equation 2 u = f is called the Kirchhoff-Love model for the vertical deflection of a thin elastic plate. For a more elaborate history of the biharmonic problem and the relation with elasticity from an engineering point of view one may consult a survey of Meleshko [19]. In particular, the Kirchhoff-Love model with Navier conditions, u = u = 0 on, corresponds to the hinged plate model and allows rewriting these fourth order problems as a second order system. Among nonlinear problems for fourth order elliptic equations with λ = 0, there exist several results concerning to semilinear equations with power type nonlinear sources (see for example [17]). A crucial role is played by the critical (Sobolev) exponent, N+ N, which appears whenever N > (see [5]). The case λ > 0 is quite different. The singular term u is related to the Hardy inequality: Date: January 2, Mathematics Subject Classification. 35J91, 35B25. Keywords. Semilinear biharmonic problems, Hardy potential, Navier conditions, existence and nonexistence results. Authors are supported by project MTM , MEC, Spain. 1

2 2 MAYTE PÉREZ-LLANOS AND ANA PRIMO Let u C 2 () and N >, then it holds that Λ N u 2 dx u 2 dx, ( N 2 (N ) 2 ) where Λ N = is optimal (see Theorem 2.2 below). 16 First of all, it is not difficult to show that any positive supersolution of (1) is unbounded near the origin and then additional hypotheses on p are needed to ensure existence of solutions. We will say that problem (1) blows-up completely if the solutions to the truncated problems (with the λ weight + 1 instead of the Hardy type term λ ) tend to infinity for every x as n. n The main objective of this work is to explain the influence of the Hardy type term on the existence or nonexistence of solutions and to determine the threshold exponent p + (λ) to have a complete blow-up phenomenon if p p + (λ). The corresponding elliptic semilinear case with the Laplacian operator was studied in [8, 1], where the authors show the existence of a critical exponent p (λ) > 1 such that the problem has no local distributional solution if p p (λ). Furthermore, they prove the existence of solutions with p < p (λ) under some suitable hypothesis on the datum. In fourth order problems, Navier boundary conditions play an important role to prove existence results. The problem can be rewritten as a second order system with Dirichlet boundary conditions. By classical elliptic theory, we easily prove a Maximum Principle. As a consequence, we deduce a Comparison Principle that allows us to prove the existence of solutions for p < p + (λ) as a limit of approximated problems. The paper is organized as follows. In Section 2 we briefly describe the natural functional framework for our problem and the embeddings we will use throughout the paper. Section 3 is devoted to some definitions and preliminary results. First, we describe the radial solutions to the homogeneous problem that allow us to know the singularity of our supersolutions near the origin and prove nonexistence results with local arguments. The notion of solution we are going to consider in the nonexistence results is local in nature, we just ask the regularity needed to give distributional sense to the equation. In this section, we prove a Moreau type decomposition and a Picone inequality that will be used in the nonexistence proofs and which are interesting themselves. However, to prove existence of solutions to (1) with L 1 data, we will consider the solution obtained as limit of approximations. For this, we use a comparison result that immediately follows from Navier boundary conditions. In Section we prove the existence of a threshold exponent for existence of solutions, namely, when we consider an exponent over the critical, there do not exist positive solutions even in the sense of distributions. Furthermore, in Section 5, we prove that a complete blow-up phenomenon occurs over the critical exponent. Section 6 deals with the complementary interval of powers. In this range it is shown that, under some suitable hypotheses on f, problem (1) has a positive solution.

3 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM Functional Framework We briefly describe the natural framework to treat the solutions to the problem in consideration. Let R N denote a bounded and smooth domain. We define the Sobolev space W k,p () = {u L p (), D α u L p () for all 1 α k}, endowed with the norm (2) u W k,p () = or equivalently u p dx + u W k,p () = u p + 1 α k 1 α k D α u p dx D α u p, where p denotes the usual norm in L p () for 1 p <. Taking the closure of C0 k () in W k,p () gives rise to the following Sobolev space W k,p 0 (), with the norm u W k,p 0 () = 1 α k D α u p dx equivalent to (2). In fact, using interpolation theory one can get rid of the intermediate derivatives and find that ( ) 1 u W k,p () = u p dx + D k u p p dx, defines a norm which is equivalent to (2), and similarly (3) u W k,p 0 () ( = D k u p dx see for instance [2]. ) 1 p, The following embeddings will be useful in the forthcoming arguments. Theorem 2.1 (Rellich-Kondrachov s Theorem). Let k N + and 1 p < +. Suppose that R N is a Lipschitz domain. Then, W k+j,p () W j,p () and W k,p () L q (), for all 1 q with the convention that 1 p, 1 p, Np N kp = p, Np N kp = + if kp N. Moreover, this embedding is compact if q < Np N kp. The statement of the previous theorem also holds if we replace W m,p () with its proper subspace W m,p 0 (). In particular, the natural space for our problem is V = {u W 2,2 (), such that u = 0, u = 0 on } W 2,2 () W 1,2 0 ().

4 MAYTE PÉREZ-LLANOS AND ANA PRIMO Let us briefly discuss that for certain domains, H := W 2,2 () W 1,2 0 () is a Hilbert space (so it is V as a closed subset of W 2,2 () W 1,2 0 ()), endowed with the following scalar product () u, v H = u v dx, which induces the norm (5) u H = u L2 (), equivalent to (3) with k = p = 2. Note that in (5) we ignore not only the intermediate derivatives, but also some of the highest order derivatives in a larger space than W 2,2 0 (). Trivially, for any u W 2,2 () W 1,2 0 () it holds that N ( D 2 u 2 2 ) 2 u N ( 2 ) 2 ( u = 1 N ) 2 2 u = 1 x i x j x i x i N x i x i N ( u)2. i,j=1 i=1 To show the reverse inequality we refer to Theorem 2.2 in [3], where it is shown that under certain conditions on the domain, in particular for smooth domains, there exists a positive constant, independent of u, such that u W 2,2 () C u L 2 (). From Rellich-Kondrachov s Theorem it follows that, W 2,2 () W 1,r () with r 2N N 2 = 2 (6) W 2,2 () L q () with q 2N N = 2, where the embeddings are compact with strict inequalities on r and q. There exists an extensive literature about this field. For more details and properties of higher order spaces and some applications we refer for example to the books [1, 17] and references therein. See also the classic books [2, 18]. We denote by W k,p loc () as the set of functions belonging to W k,p ( ), for any. Furthermore, we need to understand deeply the behavior of the linear operator: 2 ( ) λ ( ) : W 2,2 () W 1,2 0 () W 2,2 0 () ( N 2 (N ) 2 ) It is well known that this operator is coercive if 0 λ < Λ N, noting by Λ N =. 16 This coercivity is a direct consequence of the the optimality of the constant Λ N in the Rellich inequality, stated in the next theorem. The proof of this result is due to Rellich in [25], but see also [13] for alternative proofs. For convenience of the reader, we include here an elementary proof of this inequality (based in a proof of María J. Esteban for the Laplacian operator, see also [20]). Theorem 2.2. For any u W 2,2 () W 1,2 0 () and N >, it holds that u 2 (7) Λ N dx u 2 dx, ( N 2 (N ) 2 ) where Λ N = is optimal. 16 i=1

5 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 5 Proof. We consider the following identities that follow from integrating by parts 0 u x + λu x 2 3 = ( u) 2 dx + λ 2 u 2 u u dx + 2λ L 2 () 2 dx (8) = ( u) 2 dx + λ 2 u 2 ( ) u dx 2λ u, 2 dx = ( u) 2 dx +λ 2 u 2 u 2 u, x u dx 2λ 2 dx + λ dx. Noticing that we can write Hence, N i=1 u x i x i = N i=1 ( u, x u u dx = = (ux i ) x i Nu, ), x udx N u, x u dx (N ) u, x u (N ) u 2 dx = 2 dx. u 2 dx u 2 dx. On the other hand, thanks to Caffarelli-Kohn-Nirenberg inequalities (see [10]) with p = 2, γ = 1, u 2 ( ) 2 N u 2 2 dx 2 dx. Substituting these last two expressions in (8) we obtain ( ) 0 ( u) 2 dx + λ 2 N(N ) + 2λ u 2 dx. The polinomial p(λ) = λ 2 2λ N(N ) reaches its maximum at λ = N(N ) and (7) follows. This constant Λ N is optimal but not achieved. For literature on this topic see for instance [13, 16] and the references therein. 3. Some definitions and Preliminary results For 0 λ < Λ N, let us consider the homogenous equation, (9) 2 u λ u = 0 in RN.

6 6 MAYTE PÉREZ-LLANOS AND ANA PRIMO (10) We denote by α ± (λ) = N 2 ± 1 2 (N 2 N + 8) (N 2) 2 + λ β ± (λ) = N 2 ± 1 2 (N 2 N + 8) + (N 2) 2 + λ the four real roots which give the radial solutions α±, β± to equation (9). It follows that that β (λ) α (λ) α + (λ) β + (λ). For 0 λ < Λ N, then α [0, N 2 ) and α + ( N 2, N ] and if λ = Λ N = N 2 (N ) 2, 16 then α = α + = N. Moreover, notice that if 0 λ < Λ N, then β 2, and β + (N 2). 2 β (λ) α (λ) α + (λ) β + (λ) < >< > ( ) N β (Λ N ) (N ) (N 2) β +(Λ N ) Exponents of the radial solutions to the homogeneous problem. We also remark that β is the unique bounded solution and α is the unique singular solution that belongs to the Sobolev Space W 2,2 () for λ < Λ N. We give the notion of solution we are going to consider in the nonexistence results. Definition 3.1. We say that u L 1 loc () is a supersolution (subsolution) to { 2 u = g(u) in, u = u = 0 on, in the sense of distributions if g L 1 loc () and for all positive φ C 0 u φ dx ( ) g(u) φ dx. (), we have that If u is a supersolution and subsolution in the sense of distributions, then we say that u is a distributional solution to (1). In particular, in problem (1), we consider g(u) = (λ u + u p + cf) L 1 loc (). In the previous Definition 3.1 we just ask the regularity needed to give distributional sense to the equation. As we will prove below, Remark 3.11, if u is a supersolution to (1) in the sense of distributions, then g must satisfy a regularity condition. We will use this general framework to prove nonexistence of positive solutions u satisfying u 0 in. We note here that from Navier boundary conditions, we can straightforward deduce a Strong Maximum Principle and as a consequence, a comparison result. Lemma 3.2. Strong Maximum Principle Let us consider u to be a nontrivial supersolution to { (11) 2 u = 0 in, u = u = 0 on.

7 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 7 Then u > and u > 0 in. Proof. Considering the change of variables u = v, if u is a supersolution to (11), then v is a supersolution to { v = 0 in, (12) v = 0 on. Applying the known Strong Maximum Principle to the Laplacian operator, it immediately follows that v > 0 in and then u > 0 in. Lemma 3.3. Comparison Principle Let u and v satisfy the following 2 u 2 v in, u v on, u v on. Then, v u and v u in. Proof. It is sufficient to apply to w = u v, a supersolution to (11), the previous Strong Maximum Principle. Notice that from the associated system (12), we can also easily obtain a Weak Harnack s type inequality. Lemma 3.. Weak Harnack inequality Let u be a positive distributional supersolution to (11), then for any B R (x 0 ), there exists a positive constant C = C(θ, ρ, q, R), 0 < q < 0 < θ < ρ < 1, such that u Lq (B ρr (x 0)) C ess inf B θr (x 0) u. N N 2, Related to comparison of solutions we have the following decomposition due to Moreau, see [21]. For details of the proof considering more general operators, we refer for instance to [15, 17]. However, for completeness, we perform the proof in the case of the bilaplacian operator in the Hilbert Space H = W 2,2 () W 1,2 0 (), with the scalar product defined in (), see [1, 17]. Proposition 3.5. Let u H. Then, there exist unique u 1, u 2 H such that u = u 1 +u 2, satisfying u 1 0, and u 2 0 in and u 1 u 2 dx = 0. Proof. The arguments of this proof are based on the fact that W 2,2 () W 1,2 0 () is a Hilbert Space endowed with the following scalar product u, v H = u v dx. Then, if we consider the cone of the a.e positive functions defined in, C = {v H, such that v(x) 0 a.e in } the corresponding dual cone with respect to the scalar product above is defined as C = {w H, such that u wdx 0, for every u C}. Let us take as u 1 the orthogonal projection of u H on C, namely, let u 1 be such that (u u 1 ) 2 dx = min (u ν) 2 dx. ν C

8 8 MAYTE PÉREZ-LLANOS AND ANA PRIMO Letting u 2 = u u 1, for all t 0 and ν C, it holds that (u u 1 ) 2 dx (u (u 1 tν)) 2 dx = (u u 1 ) 2 dx 2t (u u 1 ) νdx + t 2 Therefore, (13) 2t B R (0) (u u 1 ) νdx t 2 B R (0) ν 2 dx. Choosing t > 0, simplifying and making then t 0, we obtain that u 2 νdx 0, for any ν C, hence u 2 C. In particular, we can put ν = u 1, thus u 1 u 2 dx 0. ν 2 dx. Arguing analogously, for some t [ 1, 0) in (13), and letting t tend to 0, we get the reverse inequality, and hence u 1 u 2 dx = 0. Let us show next the uniqueness. Assume that u = u 1 + u 2 = v 1 + v 2 with u 1, v 1 C and u 2, v 2 C. Then 0 = u 1 v 1 + u 2 v 2, u 1 v 1 + u 2 v 2 H = u 1 v 1 H + u 2 v 2 H + 2 u 1 v 1, u 2 v 2 H = u 1 v 1 H + u 2 v 2 H 2 u 1, v 2 H 2 v 1, u 2 H u 1 v 1 H + u 2 v 2 H. This implies that u 1 = v 1 and u 2 = v 2 as desired. To conclude the proof we show that every function w C is non positive and, in particular, u 2 0. For every arbitrary nonnegative h C 0 (), consider the solution to the following problem { 2 v = h in, v = 0, v = 0 on. By the Maximum Principle v C. But then, 0 v wdx = hw dx, for every nonnegative function h C0 (). By density, we conclude that w(x) 0 a. e. x as we wanted to prove. To show existence of solutions to (1) with L 1 data, we will consider the solution obtained as limit of approximations (see [6, 7, 11, 23] for the Laplacian operator case).

9 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 9 We denote T k (s) = { s, s k k sign(s), s k, the usual truncation operator. Consider f L 1 (), f 0, and let {f n } be a sequence of functions such that, f n (x) f(x), x, and f n f in L 1 (). Let u n W 2,2 () W 1,2 0 () L () be the weak solution to problem, (1) Then u n verifies ( u n )( φ) dx = 2 u n = f n in, u n > 0 in, u n = u n = 0 on. f n φ dx, φ W 2,2 () W 1,2 0 (). Noting by u n = v n such that v n = f n, we obtain that {v n } is bounded in W 1,q 0 (), for any 1 q < N N 1. Furthermore, for all k > 0, {T k(v n )} is bounded in W 1,2 0 () (see [6]). Consequently, up a subsequence, there exists v W 1,q 0 (), for all 1 q < N, such that N 1 v n v, weakly in W 1,q N 0 (), q [1, N 1 ). With these previous properties one can prove that v n v a.e. in. Therefore by Fatou s and Vitali s Theorems it follows that v n v strongly in W 1,q N 0 (), q [1, N 1 ) and T k(v n ) T k (v) strongly in W 1,2 0 (), for all k > 0. Hence v is a positive solution of the problem v = f. Taking into account that u n = v n and applying Embedding Theorem 2.1, we conclude that there exists u W 2,q (), for all 1 q <, with v = u, such that N N 1 u n u strongly in W 2,q (), q [1, N N 1 ). By the Strong Maximum Principle for the truncated problems (1), we find that for f 0, then u n > 0 and u > 0. This shows that u is a positive solution of the problem 2 u = f in, u > 0 in, u = u = 0 on. The positive supersolution obtained above allows us to construct a positive solution to (1). This is the core of the following lemma. Lemma 3.6. Let ū L 1 loc ( ) be a positive distributional supersolution to (1) with λ Λ N, f L 1 (), f 0, and, such that ū 0 in the distributional sense. Then, there exists a positive minimal solution v L 1 () L p () to problem (1) obtained as limit of approximations. Proof. Let ū be a positive supersolution to (1) with λ Λ N. For f n = T n (f), we construct recursively a sequence {v n } L 1 () L p (),

10 10 MAYTE PÉREZ-LLANOS AND ANA PRIMO starting with 2 v 1 = f 1 in, v 1 > 0 in, v 1 = v 1 = 0 on. By the comparison principle in Lemma 3.3, it follows that v 1 ū in. By iteration, we define for n > 1, 2 v n = λ v n v p n 1 + f n 1 in, n (15) v n > 0 in, v n = v n = 0 on. As above, it follows that 0 v 1... v n 1 v n ū in, so we obtain the pointwise limit, v(x) = lim n v n (x), which verifies that 0 v ū and 2 v = λ v + vp + f in, v > 0 in, v = v = 0 on. Moreover, v has the regularity of a solution obtained as limit of approximations to (1) in. We state some inequalities we will use along the paper. We start formulating an extension of a well-known Picone identity, that in the case of regular functions and the Laplacian operator was obtained by Picone in [2] (see also [] and [1] for an extension to positive Radon measures and the p-laplacian with p > 1). Lemma 3.7. Let u, v C 2 () satisfy v > 0 and v 0 in. The following functionals, ( L(u, v) = u u ) 2 v v, ( ) u R(u, v) = u 2 2 v, v verify that R(u, v) L(u, v) 0 in. Proof. Notice that, ( ) u R(u, v) = u 2 2 v v = u 2 2 u ( u2 u u v + v v 2 v 2 2 v v ( = u u ) 2 v v 2 v v ( u u v v ) 2 L(u, v) 0. ) u + 2 u ( ) u v v v v Lemma 3.8. Let v W 2,2 () be such that v δ > 0 in. Then for all u C0 (), u 2 2 v dx v u2 dx.

11 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 11 Proof. Since v W 2,2 () and v δ > 0 in, there exists a family of regular functions v n such that v n v in W 2,2 (), v n C 2 (), v n v, a.e, and v n > δ in. 2 Furthermore, 2 is a continuous operator from W 2,2 () to W 2,2 (), see [17]. Hence 2 v n 2 v in W 2,2 (). By the previous lemma applied to v n, we deduce ( ) u (16) u 2 2 v n. Thanks to (16), integrating by parts, 2 v n u 2 dx = v n v n ( ) u 2 v n dx u 2 dx. v n Using the hypothesis on v n and Lebesgue dominated convergence theorem, we obtain 2 v v u2 dx u 2 dx, u C0 (), and the result follows. Theorem 3.9. Picone s inequality. Consider u, v W 2,2 () W 1,2 0 () such that 2 v = µ, where µ is a positive bounded Radon measure, v = v = 0 on and v 0. Then, u 2 dx 2 v v u2 dx. Proof. The Strong Maximum principle yields that v > 0 in. Set ˆv m (x) = v(x) + 1 m, m N. Note that 2ˆv m = 2 v and ˆv m v in W 2,2 () and almost everywhere. The general result follows from the previous lemma and a density argument. Namely, let u n C0 () be such that u n u in W 2,2 () W 1,2 0 (). By Lemma 3.8, it holds that 2 v ˆv m u 2 n dx = 2ˆv m u 2 n dx ˆv m u n 2 dx. Taking limits m, n and applying Fatou s Lemma we obtain the desired result. In the following result we find some estimates on the supersolutions to our problem (1). They will be used in the next section to show nonexistence of solutions when p is above the threshold p + (λ). Lemma Assume that u is a nonnegative function defined in such that u 0, u L 1 loc () and u L1 loc (). If u satisfies u 0, 2 u λ u 0 in the sense of distributions, then there exists a small ball B η (0) such that u ĉ α and u c (α +2) in B η (0), where ĉ = ĉ(η), c = c (η) and α is defined in (10). Proof. Since u is a nonnegative function satisfying u 0 in the sense of distributions and λ > 0, using the Strong Maximum Principle 3.2, it is not difficult to obtain that u δ > 0 and u ν > 0 in a small ball B η ().

12 12 MAYTE PÉREZ-LLANOS AND ANA PRIMO Fixed η > 0, in order to get the above estimate, let v W 2,2 (B η (0)) be the unique solution to: (17) 2 v λ v = 0 in, v > 0 in B η (0), v = δ, v = ν, on B η (0). By a direct computation we obtain that v(r) = C 1 r β + C 2 r α with C 1 and C 2 satisfying { C1 η β + C 2 η α = δ, C 1 β (N 2 β )η β 2 + C 2 α (N 2 α )η α 2 = ν. Since u is a positive supersolution to problem (17) and using a Comparison Principle, we conclude that u v and u v in B η (0), so u ĉ α in B η (0), and u c (α +2) in B η (0). We search a solution u = A α of the associated radial elliptic problem in B R (0), Hence by a direct computation, we obtain that 2 u λ u = up in B R (0). α ( α 2α 3 (N ) + (N 2 10N + 20)α 2 + 2(N 2)(N )α λ ) = A p 1 pα in B R (0). Therefore, α = p 1 and Ap 1 = α 2α 3 (N ) + (N 2 10N + 20)α 2 + 2(N 2)(N )α λ. It is clear that α 2α 3 (N ) + (N 2 10N + 20)α 2 + 2(N 2)(N )α λ > 0 if and only if: α < α < α +, which means p (λ) < p < p + (λ) with p + (λ) = 1+ and p (λ) = 1+. α α + α > β + which implies p < p (λ). We will see that if we perturb the bilaplacian operator with the Hardy potential, the critical power is p + (λ). Some properties of p (λ) and p + (λ) are, p + (λ) N + N as λ Λ N, p + (λ) as λ 0, p (λ) N + N as λ Λ N, p (λ) N N as λ 0.

13 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 13 p + (λ) λ p (λ) Λ N It is clear that p + (λ) and p (λ) are respectively decreasing and increasing functions on λ and then, 1 < p (λ) 2 1 p + (λ). Recall that W 2,2 () L q () with q 2N N = 2. Remark Notice that if w satisfies 2 w λ w g, w 0, in the sense of distributions in, with g L 1 loc ( ), g(x) 0,, and λ Λ N, then g must satisfy g α L 1 loc (). Indeed, we can construct a sequence {ϕ n } as follows. Take ϕ 1 such that 2 ϕ 1 = 1 in, ϕ 1 > 0 in and ϕ 1 = ϕ 1 = 0 on. The rest of the sequence is determined by 2 ϕ n ϕ n+1 = λ in, n ϕ n+1 > 0 in, ϕ n+1 = ϕ n+1 = 0 on, such that ϕ n ϕ n+1 ϕ, where ϕ is the positive solution to 2 ϕ λ ϕ = 1 in, with ϕ = ϕ = 0 on. Thanks to Lemma 3.10 there exists a positive constant c and a small ball B R (0) such that ϕ(x) c α in B R (0), where α is defined in (10). As in Lemma 3.6, we can construct a minimal solution to problem 2 w λ w = g in, w > 0 in, w = w = 0 on, obtained as limit of approximations. Then we observe that ( ) > wdx = w 2 ϕ n ϕ n+1 λ + 1 dx n ( ϕ n 2 w λ w ) dx = gϕ n dx B R (0) gϕ n dx.

14 1 MAYTE PÉREZ-LLANOS AND ANA PRIMO Therefore, {gϕ n } is an increasing sequence uniformly bounded in L 1 loc (). The result follows by the Monotone Convergence Theorem and c α g dx gϕ dx <. B R (0) B R (0) Since we are considering positive solutions to problem (1), then by setting g = u p + cf, we obtain that (u p + cf) α dx < for all B R (0). B R (0) This necessary condition will be useful in the forthcoming arguments.. Nonexistence results We devote this section to show that p + (λ) is the threshold exponent for existence of solutions, namely, when we consider an exponent over the critical, there do not exist positive solutions even in the sense of distributions. To this end we argue ad contrarium, we assume that there exists a positive supersolution to (1) and this will yield a contradiction with Hardy inequality. Theorem.1. If λ > Λ N and p > 1, or 0 < λ Λ N and p p + (λ), then the problem (1) has no positive distributional supersolution u satisfying u 0 in the sense of distributions. In case that f 0, the unique nonnegative distributional supersolution is u 0. Proof. Without loss of generality, we can assume f L (). Arguing by contradiction, we suppose that ũ is a positive supersolution satisfying ũ 0 in the sense of distributions. By Lemma 3.6, the problem (1) has a minimal solution u obtained as a limit of solutions u n of truncated problems (15). We consider some different cases. Case λ > Λ N. It suffices to consider ũ as a positive distributional supersolution to the problem 2 v v Λ N = (λ Λ N ) v + g in, v > 0 in, v = v = 0 on, where g(x) = v p + cf. By Remark 3.11, necessarily ( (λ Λ N,s ) ũ ) + g N 2 L 1 (B r (0)) for all B r (0). In particular, this implies and hence, applying Lemma 3.10, (λ Λ N,s ) ũ N 2 L 1 (B r (0)), (λ Λ N,s ) N L 1 (B r (0)), which is a contradiction. Therefore, there does not exist a positive supersolution if λ > Λ N,s. Case 0 < λ Λ N and p > p + (λ).

15 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 15 Notice that the minimal positive solution u satisfies u 0 and 2 u λ u 0 in D (). Therefore, by Lemma 3.10, there exists a positive constant C and a small ball B R (0) such that u(x) C α in B R (0), where α is defined in (10). Let us consider the positive solutions u n to the truncated problems (15). Since u n > 0 in, u n 0 in, using φ 2 u n (15), and applying Picone s inequality, u p n 1 φ 2 dx u n B R (0) with φ C 0 (B R (0)) as a test function in the approximated problems B R (0) 2 φ 2 u n dx φ 2 dx. u n B R (0) Passing to the limit as n, by Fatou s Lemma and thanks to Lemma 3.10, we deduce that φ 2 C dx φ 2 dx. (p 1)α B R (0) B R (0) Since p > p + (λ), then (p 1)α > and we obtain a contradiction with the Rellich inequality. Case λ < Λ N and p = p + (λ). This case is more delicate because we consider the threshold exponent. In the sequel, the constant C may vary from line to line. Since the mimimal positive solution u satisfies u 0 and 2 u λ u 0 in D (), then thanks to Lemma 3.10, there exists η small enough and ĉ = ĉ(η) such that u(x) ĉ α in B η (0), where α is defined in (10). Without loss of generality, from now on, we can fix η < e 1. Next we define w = A a (log ( )) b ( ( )) b 1 1 in B η (0), with A = log η where a = α, and b (0, 1) will be chosen below conveniently small. Since λ < Λ N, then w W 2,2 (B η (0)). By a direct but tedious computation, we get 2 w λ w ( ( b ( ( b 3 = Ab 1 log )) a 1 C 1 (a, b, N) + Ab log )) a C 2 (a, b, N) ( ( b 2 ( ( b 1 1 +Ab log )) a 1 C 3 (a, b, N) + Ab log )) a C (a, b, N) ( ( b 1 +A log )) { a a 2a 3 (N ) + a 2 (N 2 10N + 20) + 2a(N 2)(N ) λ }. Notice that a 2a 3 (N ) + a 2 (N 2 10N + 20) + 2a(N 2)(N ) λ = 0, thus the term of order (log(1/)) b a cancels. Then, since we are taking η < e 1, the leader term has order (log(1/)) b 1 a, so one can conclude that (18) 2 w λ w AbC ( log ( 1 )) b 1 a in B η (0), for some C = C(a, b, N) small if b 1. On the other hand, we observe that ( )) bp+(λ) 1 w p+(λ) = A p+(λ) (log p+(λ)a in B η (0).

16 16 MAYTE PÉREZ-LLANOS AND ANA PRIMO Consider the function ( )) p+(λ)b 1 h(x) = A (log (p+(λ) 1). Since < e 1 in B η (0), we have that h > 0 and h L (B η (0)) being ( ) b 1 h = η α (p+(α) 1) log. η Furthermore, h satisfies and then thanks to (18), hw p+(λ) = A a, (19) 2 w λ w Cbhwp+(λ) in B η (0). Let us construct one supersolution of (19). Define u 1 = c 1 u. Observe that w = η α on B η (0). It is easy to check that there exists C > 0 such that w C η (α+2) on B η (0). It immediately follows that u 1 c 1 ĉw on B η (0) and u 1 c 1c C ( w) on B η (0). Noticing C 2 = min{ĉ, c C }, it is sufficient to take c 1 C 2 > 1 ensuring that u 1 w on B η (0) and u 1 w on B η (0). We claim that u 1 w in B η (0). Indeed, we can choose b sufficiently small such that c 1 p+(λ) 1 b h L (), so that 2 u 1 λ u 1 c1 p+(λ) 1 u p+(λ) 1 bhu p+(λ) 1 in B η (0). Define by ϖ = w u 1. This function verifies 2 ϖ λ ϖ Cbh ( w p+(λ) u p+(λ) Now we need to use Proposition 3.5, a type Moreau decomposition. θ = ϖ + ϕ, with ϕ satisfying 2 ϕ = 0 in B η (0), ϕ 0 in B η (0), ϕ = ϖ on B η (0), ϕ = ϖ on B η (0). 1 ). To this end we define Since ϖ 0 on B η (0) and ϖ 0 on B η (0), it follows that θ ϖ and θ W 2,2 (B η (0)) W 1,2 0 (B η (0)) satisfies (20) { ( 2 θ λ θ Cbh θ = θ = 0 on B η (0). w p+(λ) u p+(λ) 1 It suffices to show that θ 0 in B η (0). Thanks to Proposition 3.5, we can decompose it as θ = v 1 + v 2 being v 1, v 2 W 2,2 (B η (0)) W 1,2 0 (B η (0)) such that v 1 0 and v 2 0. Taking v 1 as a test function in (20) yields θ v 1 dx λ θv 1 dx ), bh(w p+(λ) u p+(λ) 1 )v 1 dx.

17 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 17 Taking into account that v 1 v 2 dx = 0, we deduce that v 1 2 θv 1 dx λ dx bh(w p+(λ) u p+(λ) 1 )v 1 dx. Using now that v 2 0 and θ v 1, together with Rellich inequality in the left hand side, it follows that (21) v 1 2 dx c λ bh(w p+(λ) u p+(λ) 1 )v 1 dx, where c λ = Λ N Λ N λ > 0 (recall that λ < Λ N ). Then by convexity, the fact that ϖ θ v 1 and (21), we obtain that v 1 2 dx c λ p + (λ)bhw p+(λ) 1 v1dx. 2 Notice that taking b enough small, there exists 0 < ɛ < 1 such that hw p+(λ) 1 ɛλ N c λ p + (λ)b in B η (0), so applying again Rellich inequality, we get that v 1 2 dx c λ p + (λ)bhw p+(λ) 1 v1dx 2 ɛ v 1 2 dx. It implies that v 1 W 2,2 () W 1,2 0 (B = 0, hence v η(0)) 1 = 0, and then θ = v 2 0 as we wanted to show. Once we have that u 1 w in B η (0) we reach the desired contradiction just observing that, as before, u p+(λ) 1 1 φ 2 dx φ 2 dx. But then ( ( )) 1 (p+(λ) 1)b φ 2 log dx φ 2 dx, which contradicts the optimality of the constant in Rellich inequality. Case p = p + (λ) and λ = Λ N. In this case we have that α = N and p + (Λ N ) = N + 2 N. If ũ is a supersolution to (1), then thanks to Lemma 3.10 and Remark 3.11, we have that C p α (p+(λ N )+1) dx α ũ p+(λ N ) dx <. Since α (p + (Λ N ) + 1) = N, we reach the desired contradiction.

18 18 MAYTE PÉREZ-LLANOS AND ANA PRIMO 5. Complete blow up results The nonexistence result obtained above for p p + (λ) is very strong in the sense that a complete blow-up phenomenon occurs in two different senses. a) If u n is the solution to the approximated problem with p p + (λ), where the Hardy potential is substituted by the bounded weight ( + 1 n ) 1, then u n (x) as n. b) If u n is the solution to the approximated problem with p = p n < p + (λ) and p n p + (λ) as n, then u n (x) as n Blow up for the approximated problems when λ > Λ N and p > 1, or 0 < λ Λ N and p p + (λ). We get a blow-up behavior for the following approximated problems. Theorem 5.1. Let u n L 1 () L p () be a positive solution to the problem 2 u p n u n = (22) λa n (x)u n + c f in, n up n u n = u n = 0 on, 1 with f 0, and a n (x) = + 1. We consider λ > Λ N and p > 1, or 0 < λ Λ N and p p + (λ). n Then u n (x 0 ), x 0. Proof. Without loss of generality, we can assume that f L (). The existence of a positive solution to problem (22) follows using classical sub-supersolution arguments. Thanks to the monotonicity of the nonlinear term and the coefficient a n we can assume the existence of minimal solution u n such that u n u n+1 for all n 1. Therefore to get the blow-up result we have just to show the complete blow-up for the family of minimal solutions denoted by u n. Applying weak Harnack inequality in Lemma 3., we conclude that for any B R (x 0 ), there exists a positive constant C = C(θ, ρ, R), which may vary from line to line, with 0 < θ < ρ < 1, such that (23) u n L1 (B ρr (x 0)) C ess inf B θr (x 0) u n Cu n (x 0 ) C. We claim that, in particular, there exists r > 0 and a positive constant C = C(r), such that u n dx C, uniformly in n N. If 0 B ρr (x 0 ), the claim follows directly by (23) for any B r (0) B ρr (x 0 ). If 0 B ρr (x 0 ), let us consider a smooth curve Γ joining x 0 with the origin. We define r = 1 2 min dist(x, ), x Γ to ensure that B r (x), for every x Γ. Applying Weak Harnack inequality, we get (2) u n dx Cu n (x 0 ) C. B r(x 0) Therefore, thanks to the uniform integral estimate (2), we conclude that D B r (x 0 ), inf D u n 1 u n (x) dx 1 u n (x) dx C D D D. D B r(x 0)

19 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM. 19 Now, we take x 1 B r (x 0 ) Γ and D 1 = B r (x 0 ) B θr (x 1 ), with 0 < θ < 1. Once more, Weak Harnack inequality yields, u n dx C ess inf u n ess inf u n C B rθ (x 1) D 1 D 1. B r(x 1) Recursively, in a finite number of steps, we conclude the claim. Furthermore, by the monotone convergence theorem, there exists u 0 such that u n u strongly in L 1 (B r (0)). Let us take ϕ, the solution to the problem { 2 ϕ = χ Br(0) in, ϕ = ϕ = 0 on, as a test function in (22). Then we have C u p n u n (x) dx = ϕ dx + λ a n (x)u n ϕ dx + c fϕ dx. n up n By the monotone convergence theorem and Fatou s Lemma, u p n u p in L 1 n up loc (), n a n (x)u n u in L1 loc (). Thus it follows that u is a very weak supersolution to (1) in B r1 (0) B r (0), a contradiction with Theorem Blow up when p n p + (λ). We prove now another strong blow-up result when the power p n p + (λ). Theorem 5.2. Assume that p n satisfies p n < p + (λ) and p n p + (λ) as n and f 0. Let u n L 1 loc () be a very weak supersolution to the problem { 2 u n λ u n (25) + upn n + f in, u n = u n = 0 on. Then u n (x 0 ), x 0. Proof. Without loss of generality we can assume that f L (). Suppose by contradiction that there exists a subsequence denoted by p n and a supersolution u n such that for some point x 0 we have u n (x 0 ) C 0 <, n N. It is not restrictive taking p n (λ) = 1 + α + 1. n As before, thanks to the weak Harnack inequality, we can deduce the existence of r > 0 and a positive constant C = C(r) such that u n (x) dx C, uniformly in n N. If u n L 1 loc () is a supersolution to problem (25), then it can be shown as in Lemma 3.6 that there exists a minimal solution to (25) in 1 with 0 1 obtained by approximation.

20 20 MAYTE PÉREZ-LLANOS AND ANA PRIMO Let us denote by v n u n this minimal solution. Then v n solves { 2 v n = λ v n (26) + vpn n + f in 1, v n = v n = 0 on 1. Take φ, the solution to the problem { 2 φ = 1 in 1, φ = φ = 0 on 1, as a test function in (26). Since v n u n, there results that C v n (x) dx = 1 g n (x)φdx, 1 where g n (x) = λ v n + vpn n + f. Thus, 1 g n φ dx C for all n. This implies that g n is uniformly bounded in L 1 loc ( 1) and then g n µ in the sense of measures. Using the usual change of variables 2 v n = ( )( v n ) = w n in (26) we get, (27) { ( )wn = λ v n + vpn n + f in 1, v n = u n = 0 on 1. Now take T k (w n ) ϕ with ϕ C0 ( 1 ) as a test function in (27). Invoking the previous boundedness, we get T k (w n ) 2 ϕdx + 1 Θ k (w n )( ϕ)dx = 1 g n (x)ϕt k (w n )dx k 1 g n (x)ϕdx C, 1 where Θ k (s) = s 0 T k (σ)dσ. Hence, T k (w n ) T k (w) weakly in W 1,2 loc (). Applying the renormalized theory, the Embedding Theorem 2.1, and the fact that v n = w n, we conclude that there exists a non-negative function v such that v n v strongly in L q loc (), q < N N 3. Let ψ C0 (B r (0)) be a nonnegative function. By Fatou Lemma it follows that lim g n (x)ψdx v p+(λ) v ψ ψdx + λ n dx + fψdx. Therefore, using ψ as a test function in (26) and passing to the limit as n, we conclude that v 2 ψ dx v p+(λ) v ψ ψdx + λ dx + fψdx. Hence v is a positive distributional supersolution to (1) and then we reach a contradiction.

21 SEMILINEAR BIHARMONIC PROBLEMS WITH A SINGULAR TERM Existence of solutions: p < p + (λ) The goal of this section is to prove that, under some suitable hypotheses on f, problem (1) has a positive solution, if we consider the complementary interval of powers, namely, 1 < p < p + (λ). For the existence result, we first analyze the case f 0. We distinguish some cases: (A) 0 < λ < Λ N. (I) If 1 < p < N + and is a bounded domain, there exists a positive solution to N problem (1) with f 0 using the classical Mountain Pass Theorem in the Sobolev space W 1,2 0 () W 2,2 () (see for example [5, 17, 22]). (II) If N + N < p < p +(λ), there exists a positive solution in R N. By setting w(x) = A α with α = p 1 and Ap 1 = α 2α 3 (N ) + (N 2 10N + 20)α 2 + 2(N 2)(N )α λ, then w is a supersolution to (1) in any domain. It is clear that if p (λ) < p < p + (λ), then A p 1 > 0. Notice that w L p w loc (RN ), L1 loc (RN ) and the solution could be verified in distributional sense. This is the way in which the critical powers p (λ) and p + (λ) appear. (B) If λ = Λ N, then p + (Λ N ) = N + N = p (Λ N ). Define the Hilbert space H() as the completion of C0 () with respect to the norm φ 2 = ( φ 2 φ 2 Λ N )dx. Invoking the improved Hardy-Sobolev inequality (see for example [9, 16]), then classical variational methods in the space H() allow us to prove the existence of a positive solution w to problem (1) with f 0. Remark 6.1. In the presence of a source term f 0, if f(x) c0 with c 0 > 0 and sufficiently small, by similar arguments as above we can show the existence of a supersolution. Then, the existence of a minimal solution to problem (1) follows for all p < p + (λ). Aknowledgements. The authors would like to thank I. Peral for the motivation to study this problem and B. Abdellaoui for helpful comments and discussions. References [1] Abdellaoui, B. and Peral, I. Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian with a critical potential. Ann. di Mat. Pura e Applicata 182, (2003) [2] Adams, R. A.Sobolev spaces. Pure and Applied Mathematics, 65. Academic Press, New York-London, [3] Adolfsson, V. L 2 -integrability of second-order derivatives for Poisson s equation in nonsmooth domains. Math. Scand. 70 (1992), (1), [] Allegretto, W. and Huang, Y. X. A Picone s identity for the p Laplacian and applications, Nonlinear Ana. T.M.P (1998), [5] Bernis, F., García Azorero, J. and Peral, I. Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differential Equations 1 (2), (1996),

22 22 MAYTE PÉREZ-LLANOS AND ANA PRIMO [6] Bénilan, P. Boccardo, L. Gallouët, T. Gariepy, R. Pierre, M. and Vazquez, J.L. An L 1 -theory of existence and uniqueness of solutions of nonlinear elliptic equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. () 22, no. 2 (1995), [7] Boccardo, L. Gallouët, T. and Orsina, L. Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Lináire 13 (1996), [8] Brezis, H., Dupaigne, L. and Tesei, A. On a semilinear elliptic equation with inverse-square potential. Selecta Math. (N.S.) 11 (2005), no. 1, 1 7. [9] Brezis, H. and Vázquez, J. L. Blow-up solutions of some nonlinear elliptic problems. Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, [10] Caffarelli, L., Kohn, R. and Nirenberg, L. First Order Interpolation Inequality with Weights, Compositio Math. 53 (198), [11] Dall Aglio, A Approximated solutions of equations with L 1 data. Application to the H-convergence of quasi-linear parabolic equations, Ann. Mat. Pura Appl., 170 (1996), [12] Dávila, J. Singular solutions of semi-linear elliptic problems. Handbook of differential equations: stationary partial differential equations. Vol. VI, , Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, [13] Dold, W. J., Galaktionov, V., Lacey, A.A. and Vázquez, J. L. Rate of approach to a singular steady state in quasilinear reaction-diffusion equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. () 26 (1998), no., [1] Dupaigne, L.,A nonlinear elliptic PDE with the inverse square potential. J. Anal. Math. 86 (2002), [15] Gazzola, F. and Grunau, H.C. Critical dimensions and higher order Sobolev inequalities with remainder terms. NoDEA Nonlinear Differential Equations Appl. 8 (2001), (1), 35. [16] Gazzola, F., Grunau, H. C. and Mitidieri, E. Hardy inequalities with optimal constants and remainder terms. Trans. Amer. Math. Soc. 356 (200), (6), [17] Gazzola, F., Grunau, H. C. and Sweers, G. Polyharmonic boundary value problems. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. Lecture Notes in Mathematics, Springer-Verlag, Berlin, [18] Gilbarg, D. and Trudinger, N. S. Elliptic partial differential equations of second order. Second edition. Grundlehren der Mathematischen Wissenschaften, 22. Springer-Verlag, Berlin, [19] Meleshko, V. V. Selected topics in the history of the two-dimensional biharmonic problem. Applied Mechanics Reviews, 56:3385, [20] Mitidieri, A. A simple approach to Hardy inequalities. Mat. Zametki 67 (2000), (), [21] Moreau, J.J. Semi-continuité du sous-gradient d une fonctionnelle. C. R. Acad. Sci. Paris 260 (1965) [22] Xiong, H., Shen, Y. T. Nonlinear Biharmonic Equations with Critical Potential. Acta Mathematica Sinica. [23] Murat, F. Soluciones renormalizadas de EDP elipticas no lineales, Preprint 93023, Laboratoire d Analyse Numérique de l Université Paris VI (1993). [2] Picone, M. Sui valori eccezionali di un paramtro da cui dipende una equazione differenziale lineare ordinaria del secondo ordine. Ann. Scuola. Norm. Pisa. 11, (1910) 1 1. [25] Rellich, F. Halbbeschränkte Differentialoperatoren höherer Ordnung. Proceedings of the International Congress of Mathematicians, 195, Amsterdam, vol. III, , Dpto de Matemáticas, Universidad Autónoma de Madrid, Campus Cantoblanco, 2809 Madrid, Spain. addresses: mayte.perez@uam.es, ana.primo@uam.es.