TITLE

INFINITELY MANY WEAK SOLUTIONS FOR A p-LAPLACIAN EQUATION WITH NONLINEAR BOUNDARY CONDITIONS

AUTHOR(S)
Ji-Hong Zhao; Pei-Hao Zhao
PUB. DATE
April 2007
SOURCE
Electronic Journal of Differential Equations;2007, Vol. 2007, p1
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
We study the following quasilinear problem with nonlinear boundary conditions -Δpu + a(x)|u|p-2u = f(x, u) in Ω, |&∇u|p-2 힉u/힉v = g(x, u) on Δ &Omega, where Ω is a bounded domain in ℝN with smooth boundary 힉/힉v and is the outer normal derivative, Δpu = div(|∇u|p-2∇u) is the p-Laplacian with 1 < p < N. We consider the above problem under several conditions on f and g, where f and g are both Carathéodory functions. If f and g are both superlinear and subcritical with respect to u, then we prove the existence of infinitely many solutions of this problem by using "fountain theorem" and "dual fountain theorem" respectively. In the case, where g is superlinear but subcritical and f is critical with a subcritical perturbation, namely f(x, u) = |u|p∗2u + &lamda;|u|r-2u, we show that there exists at least a nontrivial solution when p < r < p∗ and there exist infinitely many solutions when 1 < r < p, by using "mountain pass theorem" and "concentration-compactness principle" respectively.
ACCESSION #
28108127

 

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