Local Ground-State and Mountain Pass Solutions for a $p$-Kirchhoff Equation with Critical Exponent

Abstract

We study a Kirchhoff-type equation where the diffusion coefficient is non-locally affected, the nonlinear diffusion phenomenon is governed by the $p$-Laplace operator and the population supply presents critical growth. The energy functional associated with the equation is not bounded from below so that there is no global ground-state; however, we prove the existence of a positive local ground-state. We also prove that the equation has a positive solution of mountain pass type. The concentration-compactness principle is a main tool in our approach.