Quasilinear Problems with the Competition Between Convex and Concave Nonlinearities and Variable Potentials
Abstract
The purpose of this paper is to prove some existence and non-existence theorems for the nonlinear elliptic problems of the form -{\Delta}_{p}u={\lambda}k(x)u^{q}\pmh(x)u^{{\sigma}} if x\in{\Omega}, subject to the Dirichlet conditions u_{1}=u_{2}=0 on \partial{\Omega}. In the proofs of our results we use the sub-super solutions method and variational arguments. Related results as obtained here have been established in [Z. Guo and Z. Zhang, W^{1,p} versus C^{1} local minimizers and multiplicity results for quasilinear elliptic equations, Journal of Mathematical Analysis and Applications, Volume 286, Issue 1, Pages 32-50, 1 October 2003.] for the case k(x)=h(x)=1. Our results reveal some interesting behavior of the solutions due to the interaction between convex-concave nonlinearities and variable potentials.
Keywords
Cite
@article{arxiv.1104.4626,
title = {Quasilinear Problems with the Competition Between Convex and Concave Nonlinearities and Variable Potentials},
author = {Dragos-Patru Covei},
journal= {arXiv preprint arXiv:1104.4626},
year = {2011}
}
Comments
10 pages