Generalized quasilinear equations with sign-changing unbounded potential

Abstract

We are interested in studying the existence of solution for the generalized quasilinear Schrödinger equation: (P) $\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\left(g^2\right(u\left)\mathrm{\nabla }u\right)+g\left(u\right)g^{\prime }\left(u\right)|\mathrm{\nabla }u{|}^2+V\left(x\right)u=h(x,u),\\ u\in H^1\left({\mathbb{R}}^N\right)\end{array}$(P) in ${\mathbb{R}}^N$, where $N\ge 3$, g and h are suitable smooth functions and V is a potential that may change sign and be unbounded. By using a change of variables and variational arguments, we obtain the existence of a nontrivial solution for the given problem.

Keywords:

• Quasilinear equations
• variational methods
• splitting

2010 Mathematics Subject Classifications:

• 35J20
• 35Q35
• 35Q40

Acknowledgments

Part of this work was done while the authors were visiting the Universidade de Brasília; they would like to thank Professor Ricardo Ruviaro and all faculty and staff of UnB for their hospitality.

Disclosure statement

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

Additional information

Funding

The second author was partially supported by FAPEMA.