Affiliation:
1. Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, P. R. China
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.
Funder
National Natural Science Foundation of China
Natural Science Foundation of Shanxi Province
Cited by
10 articles.
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