On the quasilinear elliptic problem with a Hardy-Sobolev critical exponent. (English) Zbl 1275.35112

Chen, Guanwei; Ma, Shiwang

Dyn. Partial Differ. Equ. 8, No. 3, 225-237 (2011).

Summary: 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. 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.

MSC:

35J92	Quasilinear elliptic equations with \(p\)-Laplacian
35B33	Critical exponents in context of PDEs
35B38	Critical points of functionals in context of PDEs (e.g., energy functionals)
35J20	Variational methods for second-order elliptic equations
35J25	Boundary value problems for second-order elliptic equations

Keywords:

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