Fast and slow decaying solutions for H 1 -supercritical quasilinear Schrödinger equations

We consider the following quasilinear Schrödinger equations of the form ▵u-εV(x)u+u▵u2+up=0,u>0inRNandlim|x|→∞u(x)=0,where N≥3,p>N+2N-2,ε>0 and V(x) is a positive function. By imposing appropriate conditions on V(x), we prove that, for ε=1, the existence of infinity many positive solutions with slow decaying O(|x|-2p-1) at infinity if p>N+2N-2 and, for ε sufficiently small, a positive solution with fast decaying O(|x|2-N) if N+2N-2