Quasilinear Schrödinger equations with critical exponential growth in ℝ2: Existence and concentration behavior of solutions

In this paper, we study two classes of quasilinear Schrödinger equations that describe several phenomena in the real world, such as superfluid thin films or self-channeling of a high-power ultrashort laser in matter. Combining analytical skills and the Trudinger–Moser inequality, we establish the existence of nontrivial solutions for such problems when the nonlinear reaction term satisfies subcritical and critical exponential growth, respectively. In particular, we prove the existence of ground state sign-changing solutions via the constrained variational method and quantitative deformation techniques. In this paper, we are also concerned with the asymptotic behavior of these solutions on the vanishing set of the source potential. Our results extend and complement several recent contributions in the literature.