High energy solutions of modified quasilinear fourth-order elliptic equation

This paper focuses on the following modified quasilinear fourth-order elliptic equation: {Delta(2)u - (a + b integral(R3) vertical bar del u vertical bar(2) dx) Delta u + lambda V(x)u - 1/2 Delta(u(2))u = f(x,u), in R-3, u(x) is an element of H-2 (R-3), where Delta(2) = Delta (Delta) is the biharmonic operator, a > 0, b >= 0, lambda >= 1 is a parameter, V is an element of C(R-3, R), f(x,u) is an element of C(R-3 x R, R). V(x) and f(x,u) u are both allowed to be sign-changing. Under the weaker assumption lim(vertical bar t vertical bar ->infinity) integral(t)(0)f(x,s)ds/vertical bar t vertical bar(3) = infinity uniformly in x is an element of R-3, a sequence of high energy weak solutions for the above problem are obtained.