Existence of weak solutions for quasilinear Schrodinger equations with a parameter

In this paper, we study the following quasilinear Schrodinger equation of the form -Delta(p)u + V(x)vertical bar u vertical bar(p-2)u - [Delta(p)(1 + u(2))(alpha/2)] alpha u/2(1 + u(2))((2-alpha)/2) = k(u), x is an element of R-N, where p-Laplace operator Delta(p)u = div(vertical bar del u vertical bar(p-2)del u) (1 < p <= N) and alpha >= 1 is a parameter. Under some appropriate assumptions on the potential V and the nonlinear term k, using some special techniques, we establish the existence of a nontrivial solution in C-loc(1,beta) (R-N ) (0 < beta < 1), we also show that the solution is in L-infinity (R-N ) and decays to zero at infinity when 1 < p < N.