Dynamics of Partial Differential Equations

Volume 8 (2011)

Number 3

On the quasilinear elliptic problem with a Hardy-Sobolev critical exponent

Pages: 225 – 237

DOI: http://dx.doi.org/10.4310/DPDE.2011.v8.n3.a3

Authors

Guanwei Chen (School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China)

Shiwang Ma (School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China)

Abstract

In this article, we consider a quasilinear elliptic equation involving Hardy-Sobolev critical exponents and superlinear nonlinearity. The right hand side nonlinearity f(x, u) which is (p − 1)-superlinear nearby 0. However, it does not satisfy the usual Ambrosetti-Rabinowitz condition (AR-condition). Instead we employ a more general condition. Using a variational approach based on the critical point theory and the Ekeland variational principle, we show the existence of two nontrivial positive solutions. Moreover, the obtained results extend some existing ones.

Keywords

p-Laplacian; Hardy-Sobolev critical exponent; (PS)c-condition; Mountain pass lemma; Ekeland variational principle

2010 Mathematics Subject Classification

35A15, 35K91

Full Text (PDF format)