Solutions for fourth-order Kirchhoff type elliptic equations involving concave–convex nonlinearities in RN

In this paper, we show the existence and multiplicity of solutions for the following fourth-order Kirchhoff type elliptic equations Δ 2 u − M ( ‖ ∇ u ‖ 2 2 ) Δ u + V ( x ) u = f ( x , u ) , x ∈ R N , where M ( t ) : R → R is the Kirchhoff function, f ( x , u ) = λ k ( x , u ) + h ( x , u ) , λ ≥ 0 , k ( x , u ) is of sublinear growth and h ( x , u ) satisfies some general 3-superlinear growth conditions at infinity. We show the existence of at least one solution for above equations for λ = 0 . For λ > 0 small enough, we obtain at least two nontrivial solutions. Furthermore, if f ( x , u ) is odd in u , we show that above equations possess infinitely many solutions for all λ ≥ 0 . Our theorems generalize some known results in the literatures even for λ = 0 and our proof is based on the variational methods.