Quasilinear Problems with the Competition Between Convex and Concave Nonlinearities and Variable Potentials

Details Quasilinear Problems with the Competition Between Convex and Concave Nonlinearities and Variable Potentials Quasilinear Problems with the Competition Between Convex and Concave Nonlinearities and Variable Potentials

2011-04-24

arxiv.org/abs/1104.4626v2

Mathematics

Classical Analysis and ODEs

10 pages

Scientific paper

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.

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