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Multiple solutions of a quasilinear elliptic problem involving nonlinear boundary condition on exterior domain
Authors:Caisheng Chen  Qi Zhang
Institution:College of Science, Hohai University Nanjing 210098, , Jiangsu Province, China
Abstract:In this paper, we study the multiplicity of non‐negative solutions for the quasilinear p‐Laplacian equation with the nonlinear boundary condition math image (1) where Δp denotes the p‐Laplacian operator, defined by △ pu = div( | ? u | p ? 2 ? u),1 < p < N, Ω is a smooth exterior domain in urn:x-wiley:01704214:media:mma1606:mma1606-math-0002. urn:x-wiley:01704214:media:mma1606:mma1606-math-0003 is the outward normal derivative, urn:x-wiley:01704214:media:mma1606:mma1606-math-0004. The parameters p,q,r are either math image or math image. The weight functions a(x),h(x),g(x) satisfy some suitable conditions. Using the decomposition of the Nehari manifold and the variational methods, we prove that problem (1) has at least two positive solutions provided 0 < | λ | < λ1 for some λ1. Copyright © 2012 John Wiley & Sons, Ltd.
Keywords:quasilinear elliptic problem  p‐Laplacian equation  Nehari manifold  nonlinear boundary condition
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