On a class of periodic quasilinear Schrödinger equations involving critical growth in [formula omitted

Abstract

We consider the equation − Δ u + V ( x ) u − k 2 ( Δ ( | u | 2 ) ) u = g ( x , u ) , u > 0 , x ∈ R 2 , where V : R 2 → R and g : R 2 × R → R are two continuous 1-periodic functions and k is a positive constant. Also, we assume g behaves like exp ( β | u | 4 ) as | u | → ∞ . We prove the existence of at least one weak solution u ∈ H 1 ( R 2 ) with u 2 ∈ H 1 ( R 2 ) . The mountain pass in a suitable Orlicz space together with the Trudinger–Moser inequality are employed to establish this result. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrödinger equations. 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.