Ground State Solutions for a Class of Periodic Kirchhoff-Type Equation in $${\mathbb {R}}^3$$ Involving Critical Sobolev Exponent

Abstract

This paper is concerned with the following Kirchhoff-type equation $$\begin{aligned} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2dx\right) \varDelta u+V(x)u=Q(x)u^5+ f(x,u), \quad x \in {\mathbb {R}}^{3}, \end{aligned}$$ where $$a,b>0$$ are constants, V(x), Q(x) and f(x, u) are periodic in x, and the nonlinear growth of $$u^5$$ reaches the Sobolev critical exponent since $$2^*=6$$ in dimension 3. Under some suitable assumptions on V, Q and f, using variational methods, we establish the existence of nontrivial ground state solution for the above equation. Recent results from the literature are improved and extended.