Four positive solutions of a quasilinear elliptic equation in $ R^N$

Fang-Fang Liao; Chun-Lei Tang

doi:10.3934/cpaa.2013.12.2577

Article Contents

2013, Volume 12, Issue 6: 2577-2600. Doi: 10.3934/cpaa.2013.12.2577

This issue Previous Article Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation Next Article Nonexistence of positive solutions for a system of integral equations on $R^n_+$ and applications

Four positive solutions of a quasilinear elliptic equation in $ R^N$

Fang-Fang Liao^1, and
Chun-Lei Tang^1,

1.
School of Mathematics and Statistics, Southwest University, Chongqing 400715

Received: June 30, 2012

Revised: September 30, 2012

Published: October 2013

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Abstract

This paper deals with the existence of multiple positive solutions of a quasilinear elliptic equation \begin{eqnarray} -\Delta_p u+u^{p-1} = a(x)u^{q-1}+\lambda h(x) u^{r-1}, \text{in} R^N; \\ u\geq 0, \text{ a.e. }x \in R^N;\\ u \in W^{1,p}(R^N), \end{eqnarray} where $1 < p \leq 2$, $N>p$ and $1 < r < p$ $< q < p^* ( = \frac{pN}{N-p})$. A Nehari manifold is defined by a $C^1-$functional $I$ and is decomposed into two parts. Our work is to find four positive solutions of Eq. (1) when parameter $\lambda$ is sufficiently small.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J62; 35J92.

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