Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on RNRN

In this paper, we study the following fourth-order elliptic equation of the form▵2u-▵u+V(x)u-κ2▵(u2)u=h(x,u),x∈RN,where ▵2≔▵(▵)▵2≔▵(▵) is the biharmonic operator, κ⩾0κ⩾0, N⩽6N⩽6, V∈C(RN,R)V∈C(RN,R) and h∈C(RN×R,R)h∈C(RN×R,R). Under appropriate assumptions on V(x)V(x) and h(x,u)h(x,u), some existence and multiplicity results for nontrivial solutions are obtained via variational methods.