Berestycki-Lions conditions on ground state solutions for Kirchhoff-type problems with variable potentials

Abstract

Based on the strategy introduced by Chen and Tang [Adv. Nonlinear Anal. 9, 496–515 (2020)] and some new tricks, we prove that the nonlinear problem of Kirchhoff-type −a+b∫R3|∇u|2dx△u+V(x)u=f(u), x∈R3 in H1(R3) admits two classes of ground state solutions under the general “Berestycki-Lions assumptions” on the nonlinearity f, which are almost necessary conditions, as well as some weak assumptions on the potential V. Moreover, we also give a simple minimax characterization of the ground state energy. Our results improve and extend recent theorems in several directions.