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

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 \(\begin{aligned} \Delta ^{2} u \pm \Delta _{p} u +(1+\lambda V(x))u= f(u)\quad \text {in}\ {\mathbb {R}}^{4}, \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{2}u\pm {\mathrm{\Delta }}_{p}u+(1+\lambda V(x))u=f(u)\text{in}{\mathbb{R}}^4,\end{array}$$ where \(2< p< 4,\) $2<p<4,$ \(f\in C( {\mathbb {R}}, {\mathbb {R}})\) $f\in C(\mathbb{R},\mathbb{R})$ is a nonlinearity which has subcritical or critical exponential growth at infinity and \(V\in C({\mathbb {R}}^4,{\mathbb {R}})\) $V\in C({\mathbb{R}}^4,\mathbb{R})$ is a potential that vanishes on a bounded domain \(\Omega \subset {\mathbb {R}}^4.\) $\mathrm{\Omega }\subset {\mathbb{R}}^4.$ 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 \(\Omega .\) $\mathrm{\Omega }.$