In this paper we are interested in the existence of nontrivial solutions for a class of variable exponent $p(x)$-Kirchhoff type equations. We are able to prove the existence of three solutions by using the mountain pass theorem and Ekeland’s variational principle. Moreover, when $\lambda =0$, we obtain the existence of infinite many solutions by using symmetric mountain pass theorem.

We study the existence of ground state solutions of a Schrödinger-Poisson-Slater type equation with critical growth. By using the Nehari-Pohozaev manifold, we obtain the existence of ground state solutions of this system.