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Existence of Multiple Solutions for a p-Laplacian System in ℝ N with Sign-changing Weight Functions

Published online by Cambridge University Press:  20 November 2018

Hongxue Song ,
Caisheng Chen  and
Qinglun Yan
Hongxue Song
Affiliation:
(Song, Chen) College of Science, Hohai University, Nanjing 210098, P. R. China e-mail: [email protected]
Caisheng Chen
Affiliation:
(Song, Chen) College of Science, Hohai University, Nanjing 210098, P. R. China e-mail: [email protected]
Qinglun Yan
Affiliation:
(Song, Yan) College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, P. R. China e-mail: [email protected] e-mail: [email protected]
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Abstract

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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.

Keywords

35J66Nehari manifoldquasilinear elliptic systemp-Laplacian operatorconcave and convex nonlinearities
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 2 , 01 June 2016 , pp. 417 - 434
Copyright
Copyright © Canadian Mathematical Society 2016

References

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