Existence and multiplicity of nontrivial solutions for a class of Schrödinger–Kirchhoff-type equations

In the present paper, the following Schrödinger–Kirchhoff-type problemequation(P){−(a+b∫RN|∇u|2dx)Δu+λV(x)u=f(x,u),in RN,u(x)→0as |x|→∞, is studied. When N=2,3,4N=2,3,4 and V(x)=1V(x)=1, the existence theorem of nontrivial weak solutions for problem (P) is obtained. When N=1,2,3N=1,2,3 and V(x)V(x) is more general, two existence theorems of nontrivial weak solutions and a sequence of high energy weak solutions for problem (P) are obtained.