Generalized eigenvalue problems of nonhomogeneous elliptic operators and their application

Dumitru Motreanu and Mieko Tanaka

Vol. 265 (2013), No. 1, 151–184

DOI: 10.2140/pjm.2013.265.151

Abstract

We consider the equation −÷ (a(x,|∇u|)∇u) = λ|u|^p−2u (whose special case a(x,t) = t^p−2 is the p-Laplace equation) on a bounded domain Ω ⊂ ℝ^N with C² boundary, with null boundary condition. We prove that there are λ ∈ ℝ for which the equation has a nontrivial solution. As an application, by variational methods, we present the existence of a positive solution to −÷ (a(x,|∇u|)∇u) = f(x,u) in Ω, where f is asymptotically (p−1)-linear near zero and ∞, considering the nonresonant, resonant, and doubly resonant cases. We show that, generally, the spectrum of the operator −÷ (a(x,|∇u|)∇u) on W₀^1,p(Ω) is not discrete.

Keywords

quasilinear elliptic equations, nonhomogeneous operators, nonlinear eigenvalue problems, positive solutions, mountain pass theorem

Mathematical Subject Classification 2010

Primary: 35P30, 35J62, 49R05

Milestones

Received: 19 June 2012

Accepted: 14 September 2012

Published: 28 August 2013