Existence of nontrivial solutions for a quasilinear Schrodinger-Poisson system in R3 with periodic potentials

In this paper, we study the following quasilinear Schrodinger-Poisson system in R-3 { -Delta u + V( x)u + lambda phi u = f (x, u), x epsilon R-3, -Delta phi - epsilon(4)Delta(4)phi = lambda u(2), x epsilon R-3, where lambda and epsilon are positive parameters, Delta(4)u = div(|del u|(2)del u), V is a continuous and periodic potential function with positive infimum, f (x, t) epsilon C(R-3 x R, R) is periodic with respect to x and only needs to satisfy some superquadratic growth conditions with respect to t. One nontrivial solution is obtained for lambda small enough and epsilon fixed by a combination of variational methods and truncation technique. Keywords: quasilinear Schrodinger-Poisson system, periodic potential, variational methods, truncation technique, nontrivial solution.