On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms

We study the existence and nonexistence of nonzero solutions for the following class of quasilinear Schrödinger equations:−Δu+V(x)u+κ2[Δ(u2)]u=h(u),x∈RN, where κ>0 is a parameter, V(x) is a continuous potential which is large at infinity and the nonlinearity h can be asymptotically linear or superlinear at infinity. In order to prove our existence result we have applied minimax techniques together with careful L∞-estimates. Moreover, we prove a Pohozaev identity which justifies that 2⁎=2N/(N−2) is the critical exponent for this class of problems and it is also used to show nonexistence results.