Combined effects in planar quasilinear Schrödinger equations with superlinear reaction

Abstract

In this paper, we prove the existence of nontrivial solutions for the following planar quasilinear Schrödinger equation: − Δ u + V ( x ) u − Δ ( u 2 ) u = g ( u ) , x ∈ R 2 , where V ∈ C ( R 2 , [ 0 , ∞ ) ) and g ∈ C ( R , R ) is of subcritical exponential growth satisfying some mild conditions. In particular, by means of the Trudinger–Moser inequality, we give a different method from the one of the polynomial growth nonlinearities to prove the Brézis–Lieb split property when f has subcritical exponential growth. Our result extends and complements the one of Chen–Rădulescu–Tang–Zhang (Rev. Mat. Iberoam. 36 (2020) 1549–1570) dealing with the higher dimensions N ⩾ 3 to the dimension N = 2.