We study the existence of ground state solutions for nonlinear elliptic equations involving the biharmonic operator of the form
$$
\Delta^{2}u \pm \Delta_{p}u = f(x,u) \quad \text{in } \Omega,
$$
where $\Omega \subset \mathbb{R}^N$ is a smooth domain, $\Delta^{2}$ denotes the biharmonic operator and $\Delta_p$ is the $p$-Laplacian. The nonlinear term $f(x,u)$ may exhibit logarithmic, critical Sobolev, or exponential growth, leading to serious compactness difficulties.

The problems considered include Kirchhoff--Boussinesq type equations arising in models of elastic plates and higher-order diffusion phenomena. By means of variational methods, in particular the Nehari manifold and the mountain pass theorem, we establish the existence of nontrivial ground state solutions under suitable assumptions on the nonlinearities. The results highlight the influence of higher-order operators and critical growth on the qualitative behavior of solutions.