## Topological Methods in Nonlinear Analysis

### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 29, Number 2 (2007), 265-278.

Dates
First available in Project Euclid: 13 May 2016

https://projecteuclid.org/euclid.tmna/1463148717

Mathematical Reviews number (MathSciNet)
MR2345063

Zentralblatt MATH identifier
1132.35301

#### Citation

Alves, Claudianor O.; Ding, Yanheng. Existence, multiplicity and concentration of positive solutions for a class of quasilinear problems. Topol. Methods Nonlinear Anal. 29 (2007), no. 2, 265--278. https://projecteuclid.org/euclid.tmna/1463148717

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