Two nontrivial solutions for the nonhomogenous fourth order kirchhoff equation

Abstract

In this paper, we consider the following nonhomogenous fourth order Kirchhoff equation ▵2u-(a+b∫ℝN|∇u|2dx)▵u+V(x)u=f(x,u)+g(x), in ℝN,u∈H2(ℝN) ; where ▵2u := ▵ (▵), constants a>0, b≥0a>0, b≥0.  V 2 C(ℝN;R), f 2 C(ℝN R;R) and g 2 L2(ℝN). Under more relaxed assumptions on the nonlinear term f that are much weaker than those in L. Xu and H. Chen, using some new proof techniques especially the verification of the boundedness of Palais-Smale sequence, a new result is obtained.