Fast and slow decaying solutions for H1-supercritical quasilinear Schrodinger equations

We consider the following quasilinear Schrodinger equations of the form u -eV(x) u + u u2 + u p = 0, u > 0 in RN and lim | x|.8 u(x) = 0, where N = 3, p > N+ 2 N-2, e > 0 and V(x) is a positive function. By imposing appropriate conditions on V(x), we prove that, for e = 1, the existence of infinity many positive solutions with slow decaying O(| x|-2 p-1) at infinity if p > N+ 2 N-2 and, for e sufficiently small, a positive solution with fast decaying O(| x| 2-N) if N+ 2 N-2 < p < 3N+ 2 N-2. The proofs are based on perturbative approach. To this aim, we also analyze the structure of positive solutions for the zero mass problem.