Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems

Claudianor O. Alves, Yanheng Ding

DOI: http://dx.doi.org/10.12775/TMNA.2007.011

Abstract


Using variational methods we establish existence and multiplicity
of positive solutions for the following class of quasilinear problems
$$
-\Delta_{p}u + \lambda V(x)|u|^{p-2}u= \mu
|u|^{p-2}u+|u|^{p^{*}-2}u \quad\text{in } {\mathbb R}^{N}
$$
where $\Delta_{p}u$ is the $p$-Laplacian operator, $2 \leq p < N$,
$p^{*}={pN}/(N-p)$, $\lambda, \mu \in (0, \infty)$ and
$V\colon {\mathbb R}^{N} \rightarrow {\mathbb R}$ is a continuous function
verifying some hypothesis.

Keywords


Variational methods; critical exponent; elliptic equation

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