Existence and Concentration of Solutions for a Class of Kirchhoff–Boussinesq Equation with Exponential Growth in $${\mathbb {R}}^4$$

Abstract

This paper is concerned with the existence and concentration of ground state solutions for the following class of elliptic Kirchhoff–Boussinesq type problems given by

$$\begin{aligned} \Delta ^{2} u \pm \Delta _{p} u +(1+\lambda V(x))u= f(u)\quad \text {in}\ {\mathbb {R}}^{4}, \end{aligned}$$

where \(2< p< 4,\) \(f\in C( {\mathbb {R}}, {\mathbb {R}})\) is a nonlinearity which has subcritical or critical exponential growth at infinity and \(V\in C({\mathbb {R}}^4,{\mathbb {R}})\) is a potential that vanishes on a bounded domain \(\Omega \subset {\mathbb {R}}^4.\) Using variational methods, we show the existence of ground state solutions, which concentrates on a ground state solution of a Kirchhoff–Boussinesq type equation in \(\Omega .\)