EXISTENCE OF NONTRIVIAL SOLUTIONS FOR GENERALIZED QUASILINEAR SCHRODINGER EQUATIONS WITH CRITICAL OR SUPERCRITICAL GROWTHS

In this paper, we study the following generalized quasilinear Schrodinger equations with critical or supercritical growths -div(g(2)(u)del u) + g (u)g'(u)vertical bar del u vertical bar(2) + V (x)u = f (x, u) + lambda vertical bar u vertical bar(p-2)u, x is an element of R-N, where lambda > 0, N >= 3, g : R -> R+ is a C-1 even function, g(0) = 1, g'(s) >= 0 for all s >= 0, lim(vertical bar s vertical bar ->+infinity) g(s)/vertical bar s vertical bar(alpha-1) := beta > 0 for some alpha >= 1 and (alpha - 1)g(s) > g'(s)s for all s > 0 and p >= alpha 2*. Under some suitable conditions, we prove that the equation has a nontrivial solution for small lambda > 0 using a change of variables and variational method.