Kirchhoff-Schrödinger equations in ${\mathbb{R}}^2$ with critical exponential growth and indefinite potential

Abstract

We obtain the existence of ground state solution for the nonlocal problem $$m\left({\int }_{{\mathbb{R}}^2}(|\nabla u{|}^2+b(x)u^2)\text{d}x\right)(-\Delta u+b(x)u)=A(x)f(u)\text{in}{\mathbb{R}}^2,$$ where $m$ is a Kirchhoff-type function, $b$ may be negative and noncoercive, $A$ is locally bounded and the function $f$ has critical exponential growth. We also obtain new results for the classical Schrödinger equation, namely the local case $m\equiv 1$. In the proofs we apply Variational Methods beside a new Trudinger-Moser type inequality.

22 pages