In this paper, the existence and qualitative properties of positive ground state solutions for the following class of Schrödinger equations -ε2Δu + V(x)u - ε2[Δ(u2)]u = f(u) in the whole two-dimensional space are established. We develop a variational method based on a penalization technique and Trudinger–Moser inequality, in a nonstandard Orlicz space context, to build up a one parameter family of classical ground state solutions which concentrates, as the parameter approaches zero, around some point at which the solutions will be localized. The main feature of this paper is that the nonlinearity f is allowed to enjoy the critical exponential growth and also the presence of the second order nonhomogeneous term -ε2[Δ(u2)]u which prevents us from working in a classical Sobolev space. Our analysis shows the importance of the role played by the parameter ε for which is motivated by mathematical models in physics. Schrödinger equations of this type have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.
Existence and Concentrate Behavior of Schrödinger equations with critical exponential growth in $\mathbb{R}^N$
