GROUND STATE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER-POISSON-KIRCHHOFF TYPEPROBLEMS WITH A CRITICAL NONLINEARITY IN ℝ 3

Abstract

In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following SchrödingerPoisson-Kirchhoff system  − (1+b ∫ R3 |∇u|dx)4u+V (x)u+k(x)φu=λf(x)u+|u|u, in R, −4φ=k(x)u, in R, where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ1 and λ1 is the first eigenvalue of the problem −4u + V (x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ1. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.