Quasilinear Schrodinger equations with concave and convex nonlinearities

In this paper, we consider the following quasilinear Schrodinger equation -delta u - u delta(u(2)) = k(x) |u|(q-2) u - h(x) |u|(s-2) u, u is an element of D-1,D-2(R-N),where 1 < q < 2 < s < infinity. Unlike most results in the literature, the exponent s here is allowed to be supercritical s > 2 middot 2*. By taking advantage of geometric properties of a nonlinear transformation f and a variant of Clark's theorem, we get a sequence of solutions with negative energy in a space smaller than D-1,D-2(R-N). Nonnegative solution at negative energy level is also obtained.