Abstract
Abstract We study the quasilinear Schrödinger equation izt = −∆z +W(x)z − η(|z|2)z − k[∆p(|z|2)]p′(|z|2)z in ℝ2, where W : ℝ2 →ℝ is a positive potential and the nonlinearity η : ℝ2 × ℝ → ℝ has critical or sub-critical exponential growth. Quasilinear Schrödinger equations of this type have been studied as models of several physical phenomena such as superfluid film equation, in the theory of Heisenberg ferromagnets and magnons, in dissipative quantum mechanics and in condensed matter theory. In a suitable Orlicz space together with Trudinger-Moser inequality we establish an existence of standing wave solutions for this problem. The second order nonlinearity considered in this paper corresponds to the superfluid equation in plasma physics.
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