p-laplacian problems with critical Sobolev exponents

Nonlinear Analysis 66 (2007) 454 459 www.elsevier.com/locate/na p-laplacian problems with critical Sobolev exponents Kanishka Perera a,, Elves A.B. Silva b a Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA b Departamento de Matemática, Universidade de Brasília, 70910-900, DF, Brazil Received 2 November 2005; accepted 22 November 2005 Abstract We study the existence and multiplicity of solutions of p-laplacian problems with critical Sobolev exponents using variational methods. c 2005 Elsevier Ltd. All rights reserved. MSC: primary 35J65; secondary 35B33, 47J30 Keywords: p-laplacian problems; Critical Sobolev exponent; Existence and multiplicity; Variational methods 1. Introduction We consider the quasilinear elliptic boundary value problem { p u = μ u p 2 u + λ u p 2 u + g(x, u) in Ω u = 0 on Ω (1.1) where Ω is a smooth bounded domain in R n, n 3, p u = div ( u p 2 u) is the p- Laplacian, 1 < p < n, p = np/(n p) is the critical Sobolev exponent, μ>0andλ R are parameters, and g is a Carathéodory function on Ω R satisfying g(x, t) C( t q 1 + 1), tg(x, t) pg(x, t) C(1 + t σ ) (1.2) for some 0 σ<p < q < p and { G(x, t) 0 ast 0 t p as t uniformly in x. (1.3) Corresponding author. E-mail addresses: kperera@fit.edu (K. Perera), elves@unb.br (E.A.B. Silva). URL: http://my.fit.edu/ kperera/ (K. Perera). 0362-546X/$ - see front matter c 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.11.039

K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454 459 455 Here G(x, t) = t 0 g(x, s)ds and C denotes a generic positive constant. Our goal is to study the existence and multiplicity of solutions to this problem using variational methods. The first results on critical problems for the Laplacian were obtained in a celebrated paper of Brézis and Nirenberg [3]. This pioneering work has stimulated a vast amount of research on the subject since then (see, e.g., [1,4 8,10 14,19] and their references). The main difficulty in dealing with this class of problems is the fact that the associated functional does not satisfy the Palais Smale compactness condition (PS) since the imbedding of the Sobolev space W 1,p 0 (Ω) into L p (Ω) is not compact. However, using the concentration compactness principle of Lions [15,16] it was shown in Silva and Xavier [18] that the (PS) condition holds below any fixed level if μ>0issufficiently small. Our main results are Theorem 1.1. Assume λ σ( p ), the Dirichlet spectrum of p, and G(x, t) 0 (x, t). Then there is a μ 1 > 0 such that (1.1) has a nontrivial solution for all μ (0,μ 1 ). (1.4) Theorem 1.2. Assume that g is odd in t for all x. Then there is a μ k > 0 such that (1.1) has k pairs of nontrivial solutions for all μ (0,μ k ). Corollary 1.3. If p < q < p,thereisaμ k > 0 such that the problem { p u = μ u p 2 u + λ u p 2 u + u q 2 u in Ω u = 0 on Ω has k pairs of nontrivial solutions for all μ (0,μ k ). Our theorems are related to the results of Silva and Xavier [18] and the arguments here are adapted from Perera and Szulkin [17]. In particular, the proofs make use of a new unbounded sequence of eigenvalues of p, constructed in [17] via a minimax scheme involving the cohomological index of Fadell and Rabinowitz [9], and the pseudo-index of Benci [2]. 2. Cohomological index and eigenvalues Let S denote the class of symmetric subsets of a Banach space W. Fadell and Rabinowitz [9] constructed an index theory i : S N {0, } with the following properties: (i) Definiteness: i(s) = 0 S =. (ii) Monotonicity: If there is an odd map S S,theni(S) i(s ). In particular, equality holds if S and S are homeomorphic. (iii) Subadditivity: i(s S ) i(s) + i(s ). (iv) Continuity: If S is closed, then there is a closed neighborhood N S of S such that i(n) = i(s). (v) Neighborhood of zero: If U is a bounded symmetric neighborhood of 0 in W, then i( U) = dim W. (vi) Stability: If S is closed and S Z 2 is the join of S with Z 2 = {±1}, consisting of all line segments in W R joining ±1 to points of S,theni(S Z 2 ) = i(s) + 1. (vii) Piercing property: If S, S 0, S 1 are closed and ψ : S [0, 1] S 0 S 1 is an odd mapsuchthatψ(s [0, 1]) is closed, ψ(s {0}) S 0, ψ(s {1}) S 1,then i(ψ(s [0, 1]) S 0 S 1 ) i(s). (1.5)

456 K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454 459 Let W = W 1,p 0 (Ω), normed by ( ) 1/p u = u p. Ω Dirichlet eigenvalues of p are the critical values of the C 1 functional I (u) = (2.1) 1 Ω u p, u S 1 = {u W : u =1} (2.2) by the Lagrange multiplier rule. Denote by A the class of compact symmetric subsets of S 1,let and set F l = {A A : i(a) l}, (2.3) λ l := inf max I (u). A F l u A It was shown in Perera and Szulkin [17]thatλ l are eigenvalues of p. (2.4) 3. Proof of Theorem 1.1 Solutions of (1.1) are the critical points of 1 Φ μ (u) = Ω p u p μ p u p λ p u p G(x, u), u W. (3.1) Since λ σ( p ) and the case λ<λ 1 has already been considered by Silva and Xavier [18], we assume that λ l <λ<λ l+1 for some l. Since λ>λ l,thereisana 0 F l such that I λ on A 0 and hence, by (1.4), Φ 0 (tu) t p ( 1 λ ) 0, p I (u) u A 0, t 0. (3.2) Take ϕ 0 C(CA 0, S 1 ) with ϕ 0 A0 = id, where CA 0 = (A 0 [0, 1])/(A 0 {1}) is the cone over A 0.By(1.3), Φ 0 (tu) as t (3.3) uniformly for u on the compact set ϕ 0 (CA 0 ),soφ 0 0onRϕ 0 (CA 0 ) when R > 0 is sufficiently large, and c 0 := sup u ϕ 0 (CA 0 ),t 0 Φ 0 (tu)<. (3.4) By Proposition 3.4 of Silva and Xavier [18], there is a μ 1 > 0 such that Φ μ satisfies the (PS) condition at all levels c 0 when μ (0,μ 1 ). Regarding W as a subspace of W R and CA 0 as a cone in W R, with vertex at some point W, let and A 1 = {tu : u A 0, t [0, 1]}, A = A 1 CA 0, (3.5) ϕ(v) = { Rv, v A1 Rϕ 0 (v), v CA 0, (3.6)

K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454 459 457 so that Φ μ Φ 0 0onϕ(A). On the other hand, by (1.3), Φ μ (tu) = t p ( 1 λ ) p I (u) + o(1) as t 0, u S 1, (3.7) and λ<λ l+1, so the infimum of Φ μ on B = { u S ρ : I (u/ρ) λ l+1 }, (3.8) where S ρ = { u =ρ}, is positive when 0 <ρ<r is sufficiently small. Lemma 3.1. ( A, ϕ) links B with respect to K = {tv : v CA 0, t [0, 1]}, (3.9) i.e., γ(k ) B for every map in Γ = {γ C(K, W) : γ A = ϕ}. Proof. Regarding A 0 Z 2 as a double cone over A 0,anyγ Γ can be extended to an odd map γ on K = {tv : v A 0 Z 2, t [0, 1]}, (3.10) and it suffices to show that γ( K ) B. Applying the piercing property to gives so S = A 0 Z 2, S 0 = { u ρ}, S 1 = { u ρ}, ψ(v,t) = γ(tv) (3.11) i( γ( K ) S ρ ) = i(ψ(s [0, 1]) S 0 S 1 ) i(s) = i(a 0 ) + 1 l + 1, (3.12) max I (u/ρ) λ l+1 (3.13) u γ( K ) S ρ and hence γ( K ) intersects B. Set c := inf max Φ μ(u). (3.14) γ Γ u γ(k ) Taking γ(tv) = tϕ(v) shows that c c 0 and hence Φ μ satisfies (PS) c.sincemaxφ μ (ϕ(a)) 0 < inf Φ μ (B), it follows from Lemma 3.1 that c > 0 is a critical value of Φ μ. 4. Proof of Theorem 1.2 Since g is odd in t for all x, G is even in t for all x and hence the functional Φ μ is even. We fix a μ 0 > 0 and work with μ μ 0.LetS denote the class of compact symmetric subsets of W and Γ μ the group of odd homeomorphisms γ of W such that γ = id on Φμ 1 ((, 0]). Since λ l, λ<λ l+1 when l is sufficiently large. Using (3.7), takeρ > 0 so small that inf Φ μ0 (B) >0whereB is as in (3.8), andlet iμ (A) := min i(γ (A) S ρ ), A S (4.1) γ Γ μ be the pseudo-index of Benci [2] related to i, S ρ,andγ μ.

458 K. Perera, E.A.B. Silva / Nonlinear Analysis 66 (2007) 454 459 Lemma 4.1. For any μ 0, Fm μ := { A S : iμ (A) m} m. (4.2) Proof. Take A 0 F m and, using (1.3), R >ρso large that Φ μ 0onRA 0,andlet A = {tu : u RA 0, t [0, 1]}. (4.3) Then for any γ Γ μ, γ RA0 = id and γ(0) = 0, and applying the piercing property to S = RA 0, S 0 = { u ρ}, S 1 = { u ρ}, ψ(u, t) = γ(tu) (4.4) gives i(γ (A) S ρ ) = i(ψ(s [0, 1]) S 0 S 1 ) i(s) = i(a 0 ) m. (4.5) By Lemma 4.1, c 0 := inf max Φ 0(u) <, (4.6) A Fl+k 0 u A and by Proposition 3.4 of Silva and Xavier [18], there is a 0 <μ k μ 0 such that Φ μ satisfies the (PS) condition at all levels c 0 when μ (0,μ k ).Set c m := inf max Φ μ(u), 1 m k. (4.7) A F μ u A l+m Noting that c 1 c k c 0, we will show that c 1 > 0 and hence Φ μ has k pairs of nontrivial critical points (see Benci [2]). Let A F μ l+1.takingγ = id in (4.1), i(a S ρ) l + 1, so max I (u/ρ) λ l+1 (4.8) u A S ρ and hence A B. Therefore References max Φ μ(u) inf Φ μ 0 (u) >0. (4.9) u A u B [1] A. Ambrosetti, M. Struwe, A note on the problem u = λu + u u 2 2, Manuscripta Math. 54 (4) (1986) 373 379. [2] V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (2) (1982) 533 572. [3] H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983) 437 477. [4] A. Capozzi, D. Fortunato, G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (6) (1985) 463 470. [5] G. Cerami, D. Fortunato, M. Struwe, Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (5) (1984) 341 350. [6] D.G. Costa, E.A.B. Silva, A note on problems involving critical Sobolev exponents, Differential Integral Equations 8 (3) (1995) 673 679. [7] E.A.B. Silva, S.H.M. Soares, Quasilinear Dirichlet problems in R n with critical growth, Nonlinear Anal. 43 (1) (2001) 1 20. [8] P. Drábek, Y.X. Huang, Multiplicity of positive solutions for some quasilinear elliptic equation in R N with critical Sobolev exponent, J. Differential Equations 140 (1) (1997) 106 132. [9] E.R. Fadell, P.H. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45 (2) (1978) 139 174.

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