Existence of solutions for a class of quasilinear equations with vanishing potentials

Abstract

We show the existence of positive solutions for a class of $p\mathrm{\&}q$ problems given by $\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\left(a\right(|\mathrm{\nabla }u{|}^p\left)|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\right)+V\left(x\right)b\left(|u{|}^p\right)|u{|}^{p-2}u=\mu f\left(u\right)+u^{q^{\ast }-1}\mathrm{i}\mathrm{n}{\mathbb{R}}^N,\\ u\in D^{1,p}\left({\mathbb{R}}^N\right)\cap D^{1,q}\left({\mathbb{R}}^N\right),\end{array}$where $N\ge 2$, $1<p\le q<N$, $q^{\ast }:=\frac{Nq}{N-q}$ and μ is a positive parameter. The potential V can vanish at infinity and f can be supercritical at the origin. We use the Mountain Pass Theorem and Del Pino & Felmer's arguments to overcome the lack of compactness.

Keywords:

• Critical exponent
• $p\mathrm{\&}q$-Laplacian operator
• vanishing potentials

2010 Mathematics Subject Classifications:

• Primary: 35J60
• Secondary: 35J60
• 35J10
• 35J20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Gustavo S. A. Costa was partially supported by CNPq, CAPES – Brazil.