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

#### 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.

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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