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

2011/04/24 by Dragoş-Pătru Covei, Covei, Dragos-Patru · 1 citation
Computer Science · Mathematics · #35A05 #35J20 #35J60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1104.4626

openalex publication_date 2011/04/24 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to prove some existence and non-existence\ntheorems for the nonlinear elliptic problems of the form\n-\Δ_pu=\λk(x)u^q pmh(x)u^\σ if x\∈\Ω, subject\nto the Dirichlet conditions u₁=u₂=0 on \∂\Ω. In the proofs of\nour results we use the sub-super solutions method and variational arguments.\nRelated results as obtained here have been established in [Z. Guo and Z. Zhang,\nW^1,p versus C¹ local minimizers and multiplicity results for quasilinear\nelliptic equations, Journal of Mathematical Analysis and Applications, Volume\n286, Issue 1, Pages 32-50, 1 October 2003.] for the case k(x)=h(x)=1. Our\nresults reveal some interesting behavior of the solutions due to the\ninteraction between convex-concave nonlinearities and variable potentials.\n

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