EXISTENCE OF INFINITELY MANY HIGH-ENERGY SOLUTIONS FOR A FOURTH-ORDER KIRCHHOFF-TYPE EQUATION

Abstract

We study the fourth-order elliptic equations of Kirchhoff-type Δ2u−(a+b∫ℝN|∇u|2dx)Δu+V(x)u=f(x,u)+λh(x,u),x∈ℝN, where a>0, b≥0 are constants, we have the potential V(x):ℝN→ℝ, V∈C(ℝN,ℝ). The nonlinearity λh(x,u)+f(x,u) may involve a combination of concave and convex terms. Under some suitable conditions on h,f∈C(ℝN×ℝ,ℝ) and λ∈ℝ, we show the existence of nontrivial solutions by combining the mountain pass theorem and variational methods. Moreover, we prove the existence of infinitely many high-energy solutions using the fountain theorem.