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On the Existence of Nontrivial Solutions of Quasi-asymptotically Linear Problem for the P-Laplacian

On the Existence of Nontrivial Solutions of Quasi-asymptotically Linear Problem for the P-Laplacian In this paper, we study the existence of nontrivial solutions for the following Dirichlet problem for the p-Laplacian (p>1): $$ \left\{ {\begin{array}{*{20}l} {{ - \Delta _{p} u \equiv - {\text{div}}{\left( {{\left| {\nabla u} \right|}^{{p - 2}} \nabla u} \right)} = f{\left( {x,u} \right)},} \hfill} & {{x \in \Omega ,} \hfill} \\ {{u = 0,} \hfill} & {{x \in \partial \Omega ,} \hfill} \\ \end{array} } \right. $$ where Ω is a bonded domain in ℝ N (N≥1) and f(x,u) is quasi-asymptotically linear with respect to |u| p −2 u at infinity. Recently it was proved that the above problem has a positive solution under the condition that f(x,s)/s p −1 is nondecreasing with respect to s for all x∈Ω and some others. In this paper, by improving the methods in the literature, we prove that the functional corresponding to the above problem still satisfies a weakened version of (P.S.) condition even if f(x,s)/s p −1 isn't a nondecreasing function with respect to s, and then the above problem has a nontrivial weak solution by Mountain Pass Theorem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

On the Existence of Nontrivial Solutions of Quasi-asymptotically Linear Problem for the P-Laplacian

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Publisher
Springer Journals
Copyright
Copyright © 2002 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s102550200062
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study the existence of nontrivial solutions for the following Dirichlet problem for the p-Laplacian (p>1): $$ \left\{ {\begin{array}{*{20}l} {{ - \Delta _{p} u \equiv - {\text{div}}{\left( {{\left| {\nabla u} \right|}^{{p - 2}} \nabla u} \right)} = f{\left( {x,u} \right)},} \hfill} & {{x \in \Omega ,} \hfill} \\ {{u = 0,} \hfill} & {{x \in \partial \Omega ,} \hfill} \\ \end{array} } \right. $$ where Ω is a bonded domain in ℝ N (N≥1) and f(x,u) is quasi-asymptotically linear with respect to |u| p −2 u at infinity. Recently it was proved that the above problem has a positive solution under the condition that f(x,s)/s p −1 is nondecreasing with respect to s for all x∈Ω and some others. In this paper, by improving the methods in the literature, we prove that the functional corresponding to the above problem still satisfies a weakened version of (P.S.) condition even if f(x,s)/s p −1 isn't a nondecreasing function with respect to s, and then the above problem has a nontrivial weak solution by Mountain Pass Theorem.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2002

References