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

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
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