Ground State Solutions for a Class of Problems Involving Perturbed for the Biharmonic Operator with Non-local Term

In this paper, we study the existence of ground state solutions for an equation that involves a perturbation of a biharmonic operator with a non-local term. More precisely, we study the equation \(\begin{aligned} {\mathcal {L}}u = \tau f(x,u)+\beta |u|^{2_{**}-2}u\ \text { in } \ \Omega \ \ \text { and } \ u=\frac{\partial u}{\partial \eta }=0 \ \text { on } \ \ \partial \Omega , \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{L}u=\tau f(x,u)+\beta |u|_{}^{2_{\ast \ast }^{}-2}u\text{in}\mathrm{\Omega }\text{and}u=\frac{\partial u}{\partial \eta }=0\text{on}\partial \mathrm{\Omega },\end{array}$$ where \(\Omega \subset {\mathbb {R}}^N\) $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth domain, \({\mathcal {L}}(\cdot )\) $\mathcal{L}(·)$ it is the biharmonic operator perturbed by the non-local operator that we will later define, \(\tau >0\) $\tau >0$ ; here, \( 2_{**}= \frac{2N}{N-4}\) $2_{\ast \ast }^{}=\frac{2N}{N-4}$ with \( N\ge 5\) $N\ge 5$ . We show the existence of a ground state solution using variational methods considering the subcritical case, i.e., \(\beta =0\) $\beta =0$ and the critical case, i.e., \(\beta =1\) $\beta =1$ .