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Acta Mathematicae Applicatae Sinica
, Volume 18 (4) – Jan 1, 2002

/lp/springer-journals/on-the-existence-of-nontrivial-solutions-of-quasi-asymptotically-KGoHQ3jucK

- 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

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.

Acta Mathematicae Applicatae Sinica – Springer Journals

**Published: ** Jan 1, 2002

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