Some Remarks on a Class of Nonuniformly Elliptic Equations of p -Laplacian Type
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Some Remarks on a Class of Nonuniformly Elliptic Equations of p-Laplacian Type ´ Quôc-Anh Ngô · Hoang Quoc Toan
Received: 22 April 2008 / Accepted: 4 August 2008 / Published online: 29 August 2008 © Springer Science+Business Media B.V. 2008
Abstract This paper deals with the existence of weak solutions in W01 () to a class of elliptic problems of the form − div(a(x, ∇u)) = λ1 |u|p−2 u + g (u) − h in a bounded domain of RN . Here a satisfies |a (x, ξ )| c0 h0 (x) + h1 (x) |ξ |p−1 p
for all ξ ∈ RN , a.e. x ∈ , h0 ∈ L p−1 (), h1 ∈ L1loc (), h1 (x) 1 for a.e. x in ; λ1 is the first eigenvalue for −p on with zero Dirichlet boundary condition and g, h satisfy some suitable conditions. Keywords p-Laplacian · Nonuniform · Landesman-Laser · Elliptic · Divergence form · Landesman-Laser type Mathematics Subject Classification (2000) 35J20 · 35J60 · 58E05
1 Introduction Let be a bounded domain in RN . In the present paper we study the existence of weak solutions of the following Dirichlet problem − div(a (x, ∇u)) = λ1 |u|p−2 u + g (u) − h Q.-A. Ngô () · H.Q. Toan Department of Mathematics, College of Science, Viêt Nam National University, Hanoi, Vietnam e-mail: [email protected] Q.-A. Ngô Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Singapore
(1)
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Q.-A. Ngô, H.Q. Toan
where |a(x, ξ )| c0 (h0 (x) + h1 (x)|ξ |p−1 ) for any ξ in RN and a.e. x ∈ , h0 (x) 0 and h1 (x) 1 for any x in . λ1 is the first eigenvalue for −p on with zero Dirichlet boundary condition, that is, |∇u|p dx |u|p dx = 1 . λ1 = inf 1,p u∈W0
()
Recall that λ1 is simple and positive. Moreover, there exists a unique positive eigenfunction 1,p φ1 whose norm in W0 () equals to one. Regarding the functions g, we assume that g is a p . continuous function. We also assume that h ∈ Lp () where we denote p by p−1
In the present paper, we study the case in which h0 and h1 belong to Lp () and L1loc (), respectively. The problem now may be non-uniform in sense that the functional associated to 1,p the problem may be infinity for some u in W0 (). Hence, weak solutions of the problem 1,p must be found in some suitable subspace of W0 (). To our knowledge, such equations were firstly studied by [4, 9, 10]. Our paper was motivated by the result in [2] and the generalized form of the Landesman–Lazer conditions considerred in [7, 8]. While the semilinear problem is studied in [7, 8] and the quasilinear problem is studied in [2], it turns out that a different technique allows us to use these conditions also for problem (1) and to generalize the result of [1]. In order to state our main theorem, let us introduce our hypotheses on the structure of problem (1). Assume that N 1 and p > 1. be a bounded domain in RN having C 2 boundary ∂. Consider a : RN × RN → RN , a = a(x, ξ ), as the continuous derivative with respect to ξ of ) the continuous function A : RN × RN → R, A = A(x, ξ ), that is, a(x, ξ ) = ∂A(x,ξ . Assume ∂ξ that there are a positive real
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