A Result of Multiplicity of Nontrivial Solutions to an Elliptic Kirchhoff–Boussinesq Equation Type in R N

Abstract

In this work, using Ljusternik–Schnirelman category theory, we study the existence and multiplicity of nontrivial solutions for the following class of elliptic Kirchhoff–Boussinesq type problems given by $$\begin{array}{r}{\epsilon }^4{\mathrm{\Delta }}^{2}u\pm {\epsilon }^p{\mathrm{\Delta }}_{p}u+V(x)u=f(u)+\beta |u{|}^{2_{**}-2}u\text{in }{\mathbb{R}}^N,u\in H^2({\mathbb{R}}^N),\end{array}$$ where $\epsilon >0$ , $2<p<2^{*}=2N/(N-2)$ , $2_{**}=2N/(N-4)$ and $N\ge 5$ , $V:{\mathbb{R}}^N\to \mathbb{R}$ is a continuous function and $f:\mathbb{R}\to \mathbb{R}$ is a function of $C^1$ class. We consider the subcritical case, that is, $\beta =0$ and critical case, that is, $\beta =1$ .