Quasilinear Schrödinger equations with singular and vanishing potentials involving exponential critical growth in [formula omitted

Abstract

We study the existence and nonexistence of solution for the following class of quasilinear Schrödinger equations: −Δu+V(|x|)u−[Δ(u2)]u=Q(|x|)h(u),x∈R2,u(x)→0as|x|→∞,where V and Q are potentials that can be singular at the origin, unbounded or vanishing at infinity and the nonlinearity h(s) is allowed to satisfy the exponential critical growth with respect to the Trudinger–Moser inequality. By combining variational methods in a suitable weighted Orlicz space with a version of the Trudinger–Moser inequality for this space, we obtain the existence of a nonnegative ground state solution. For this, we have used some regularity results and we need to explore a symmetric criticality type argument. Moreover, under some conditions on V, Q and h(s), we prove that this problem does not have positive radial solution. Schrödinger equations of this type have been studied as models of several physical phenomena.