Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems

We consider the following class of elliptic Kirchhoff-Boussinesq type problems given by \(\begin{aligned} \Delta ^{2} u \pm \Delta _p u = f(u) + \beta |u|^{2_{**}-2}u\ \text{ in } \ \Omega \ \ \text{ and } \ \Delta u=u=0 \ \text{ on } \ \ \partial \Omega , \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}^{2}u\pm {\mathrm{\Delta }}_{p}u=f(u)+\beta {|u|}^{2_{\ast \ast }-2}u\text{in}\mathrm{\Omega }\text{and}\mathrm{\Delta }u=u=0\text{on}\partial \mathrm{\Omega },\end{array}$$ where \(\Omega \subset \mathbb {R}^{N}\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded and smooth domain, \(2< p\le \frac{2N}{N-2}\) $2<p\le \frac{2N}{N-2}$ for \(N\ge 3\) $N\ge 3$ , \(2_{**}=\frac{2N}{N-4}\) $2_{\ast \ast }=\frac{2N}{N-4}$ if \(N\ge 5\) $N\ge 5$ , \(2_{**}=\infty \) $2_{\ast \ast }=\infty$ if \(3\le N <5\) $3\le N<5$ and f is a continuous function. We show existence and multiplicity of nontrivial solutions using minimization technique on the Nehari manifold, Mountain Pass Theorem and Genus theory. In this paper we consider the subcritical case \(\beta =0\) $\beta =0$ and the critical case \(\beta =1\) $\beta =1$ .