Magnetic Kirchhoff–Schrödinger equations with critical exponential growth and indefinite potential in R 2

Abstract

In this paper, we are concerned with the following magnetic nonlinear equation of Kirchhoff type with critical exponential growth and an indefinite potential in ${\mathbb{R}}^2$ $$m\left({\int }_{{\mathbb{R}}^2}[{|\frac{1}{i}\mathrm{\nabla }u-A(x)u|}^2+V(x)|u{|}^2]dx\right)[{(\frac{1}{i}\mathrm{\nabla }-A(x))}^{2}u+V(x)u]=B(x)f(|u{|}^2)u,$$ where $u\in H^1({\mathbb{R}}^2,\mathbb{C})$ , m is a Kirchhoff type function, $V:{\mathbb{R}}^2\to \mathbb{R}$ and $A:{\mathbb{R}}^2\to {\mathbb{R}}^2$ represent locally bounded potentials, while B denotes locally bounded and f exhibits critical exponential growth. By employing variational methods and utilizing the modified Trudinger–Moser inequality, we get ground state solutions or nontrivial solutions for the above equation. Furthermore, in the special case where m is a constant equal to 1, the equation is reduced to the following magnetic nonlinear Schrödinger equation, $${(\frac{1}{i}\mathrm{\nabla }-A(x))}^{2}u+V(x)u=B(x)f({\left|u\right|}^2)u\mathrm{in}{\mathbb{R}}^2.$$ Applying analogous methods, we can also establish the existence of ground state solutions or nontrivial solutions to this equation.