Existence and concentration behavior of solutions for a class of quasilinear elliptic equations with critical growth

Abstract

Abstract In this paper, we study a class of quasilinear elliptic equations involving the Sobolev critical exponent -\varepsilon^{p}\Delta_{p}u-\varepsilon^{p}\Delta_{p}(u^{2})u+V(x)\lvert u% \rvert^{p-2}u=h(u)+\lvert u\rvert^{2p^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where {\Delta_{p}u=\operatorname{div}(\lvert\nabla u\rvert^{p-2}\nabla u)} is the p-Laplace operator, {p^{*}=\frac{Np}{N-p}} ( {N\geq 3} , {N>p\geq 2} ) is the usual Sobolev critical exponent, the potential {V(x)} is a continuous function, and the nonlinearity {h(u)} is a nonnegative function of {C^{1}} class. Under some suitable assumptions on V and h, we establish the existence, multiplicity and concentration behavior of solutions by using combing variational methods and the theory of the Ljusternik–Schnirelman category.