On concentration of solutions for quasilinear Schrödinger equations with critical growth in the plane

Abstract

In this work, we study the existence of positive ground states and concentration phenomena for the following class of quasilinear Schrödinger equations: $-{\epsilon }^2\mathrm{div}(g^2(u)\mathrm{\nabla }u)+{\epsilon }^{2}g(u)g^{\mathrm{\prime }}(u)|\mathrm{\nabla }u{|}^2+V(x)u=Q(x)f(u)\text{in}{\mathbb{R}}^2,$ where $\epsilon >0$ is a parameter, $g:\mathbb{R}\to {\mathbb{R}}_{+}$ is a continuously differentiable function, $V(x)$ and $Q(x)$ are positive and bounded continuous potentials and the nonlinearity $f(u)$ can exhibit critical exponential growth. In order to prove our concentration result, we exploit variational arguments that take into account an interaction between the potentials V and Q, in combination with the Trudinger–Moser inequality and Moser iteration method.

Keywords:

• Quasilinear Schrödinger equation
• variational methods
• critical growth
• Trudinger–Moser inequality
• weak solutions

2020 Subject Classifications:

• 35J20
• 35J60
• 35J62

Acknowledgements

We would like to thank the referees for careful reading of the paper with many useful comments and important suggestions, which substantially helped improving the quality of the paper.

Disclosure statement

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

Additional information

Funding

Research partially supported by CNPq grant 310747/2019-8 and Grant 2019/0014 Paraíba State Research Foundation (Fapesq), Brazil.