Planar Schrödinger equations with critical exponential growth.

In this paper, we study the following quasilinear Schrödinger equation: - ε 2 Δ u + V (x) u - ε 2 Δ (u 2) u = g (u) , x ∈ R 2 , where ε > 0 is a small parameter, V ∈ C (R 2 , R) is uniformly positive and allowed to be unbounded from above, and g ∈ C (R , R) has a critical exponential growth at infinity. In the autonomous case, when ε > 0 is fixed and V (x) ≡ V 0 ∈ R + , we first present a remarkable relationship between the existence of least energy solutions and the range of V 0 without any monotonicity conditions on g. Based on some new strategies, we establish the existence and concentration of positive solutions for the above singularly perturbed problem. In particular, our approach not only permits to extend the previous results to a wider class of potentials V and source terms g, but also allows a uniform treatment of two kinds of representative nonlinearities that g has extra restrictions at infinity or near the origin, namely lim inf | t | → + ∞ t g (t) e α 0 t 4 or g (u) ≥ C q , V u q - 1 with q > 4 and C q , V > 0 is an implicit value depending on q, V and the best constant of the embedding H 1 (R 2) ⊂ L q (R 2) , considered in the existing literature. To the best of our knowledge, there have not been established any similar results, even for simpler semilinear Schrödinger equations. We believe that our approach could be adopted and modified to treat more general elliptic partial differential equations involving critical exponential growth. [ABSTRACT FROM AUTHOR]