Nonlinear perturbations of a periodic Kirchhoff–Boussinesq-type problems in \(\mathbb {R}^{N}\)

This paper is concerned with the existence of a ground state solution for the following class of elliptic Kirchhoff–Boussinesq-type problems given by \(\begin{aligned} \Delta ^{2} u \pm \Delta _{p} u +V(x)u= f(u) +\gamma |u|^{2_{**}-2}u \ \ \text{ in } \ \ \mathbb {R}^{N}, \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{2}u\pm {\mathrm{\Delta }}_{p}u+V(x)u=f(u)+\gamma {|u|}^{2_{\ast \ast }-2}u\text{in}{\mathbb{R}}^N,\end{array}$$ where \(2< p< 2^{*}= \frac{2N}{N-2}\) $2<p<2^{\ast }=\frac{2N}{N-2}$ for \( N\ge 3\) $N\ge 3$ and \(2_{**}= \infty \) $2_{\ast \ast }=\infty$ for \(N=3\) $N=3$ , \(N=4\) $N=4$ , \(2_{**}= \frac{2N}{N-4}\) $2_{\ast \ast }=\frac{2N}{N-4}$ for \(N\ge 5\) $N\ge 5$ . Here V and f are continuous functions with V being either periodic or asymptote to infinity a periodic function. The function f(u) has subcritical growth and behaves like \(|u|^{q-2}u\) ${|u|}^{q-2}u$ with \(p<q< 2_{**}\) $p<q<2_{\ast \ast }$ . We show existence of a ground state solution using variational methods considering the subcritical case, i.e, \(\gamma =0\) $\gamma =0$ and the critical case, i.e, \(\gamma =1\) $\gamma =1$ .