Quasilinear Schrödinger equations with unbounded or decaying potentials in dimension 2

Abstract

We establish the existence of nontrivial solutions for the following class of quasilinear Schrödinger equations:where 𝜅 is a positive parameter, 𝑉(|𝑥|) and 𝑄(|𝑥|) are continuous functions that can be singular at the origin, unbounded or vanishing at infinity, and the nonlinearity ℎ(𝑠) has critical exponential growth motivated by the Trudinger-Moser inequality. To prove our main result, we apply variational methods together with careful 𝐿 ∞ -estimates.