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

In the present paper, the following Schrödinger–Kirchhoff-type problem(P) { − ( a + b ∫ R N | ∇ u | 2 d x ) Δ u + λ V ( x ) u = f ( x , u ) , in R N , u ( x ) → 0 as | x | → ∞ , is studied. When N = 2 , 3 , 4 and V ( x ) = 1 , the existence theorem of nontrivial weak solutions for problem (P) is obtained. When N = 1 , 2 , 3 and 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.