Existence of Multiple Solutions for a p-Laplacian System in ℝN with Sign-changing Weight Functions. Issue 2 (1st June 2016)

• Record Type: Journal Article
• Authors: Song, Hongxue; Chen, Caisheng; Yan, Qinglun
• Is Part Of: Canadian mathematical bulletin =. Volume 59:Issue 2(2016)
• Page Start: 417
• Page End: 434
• Publication Date: 2016-06-01
• Subjects: 35J66; Nehari manifold -- quasilinear elliptic system -- p-Laplacian operator -- concave and convex nonlinearities; Mathematics -- Periodicals; Mathematics; Periodicals; 510.5
• DOI: 10.4153/CMB-2015-035-4
• Languages: English
• ISSNs: 0008-4395

Abstract

Abstract: In this paper, we consider the quasi-linear elliptic problem $$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| {{\nabla }_{u}} \right|}^{p}}dx \right)\, \text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla u \right|}^{p-2}}\nabla u \right)=\frac{\alpha }{\alpha +\beta }H\left( x \right){{\left| u \right|}^{\alpha -2}}u{{\left| v \right|}^{\beta }}+\text{ }\lambda \text{ }{{\text{h}}_{1}}\left( x \right){{\left| u \right|}^{q-2}}u, $$ $$-M\left( {{\int }_{{{\mathbb{R}}^{N}}}}{{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p}}dx \right)\, \text{div}\left( {{\left| x \right|}^{-ap}}{{\left| \nabla v \right|}^{p-2}}\nabla v \right)=\frac{\beta }{\alpha +\beta }H\left( x \right){{\left| v \right|}^{\beta -2}}v{{\left| u \right|}^{\alpha }}+\mu {{h}_{2}}\left( x \right){{\left| v \right|}^{q-2}}v, $$ $$u\left( x \right)>0, v\left( x \right)>0, x\in {{\mathbb{R}}^{N}}, $$ where $\text{ }\lambda \text{, }\mu >\text{0, }\text{1}<\text{p}<\text{N, }\text{1}<\text{q}<\text{p}<\text{p}\left( \tau +1 \right)<\alpha +\beta <{{p}^{*}}=\frac{{{N}_{p}}}{N-p}, 0\le a<\frac{N-p}{p}, a\le b<a+1, d=a+1-b>0, M\left( s \right)=k+l{{s}^{\tau }}, k>0, l, \tau \ge 0$ and the weight $H\left( x \right), \, {{h}_{1}}\left( x \right), \, {{h}_{2}}\left( x \right)$ are continuous functions that change sign in ${{\mathbb{R}}^{N}}$ . We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.