Quasilinear Schrödinger equations with singular and vanishing potentials involving exponential critical growth in R2. (July 2020) - Severo, Uberlandio B.; de Carvalho, Gilson M.
Abstract: We study the existence and nonexistence of solution for the following class of quasilinear Schrödinger equations: − Δ u + V ( | x | ) u − [ Δ ( u 2 ) ] u = Q ( | x | ) h ( u ), x ∈ R 2, u ( x ) → 0 as | 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.