Resumen de Ground States for Planar Generalized Quasilinear Schrödinger Equation with Choquard Nonlinearity

Wenting Zhao, Xianjiu Huang, Jianhua Chen, Bitao Cheng

• In this paper, we consider the following generalized quasilinear Schrödinger equation −div(g2(u)∇u) + g(u)g (u)|∇u| 2 + V(x)u = Iμ ∗ F(u) f (u), x ∈ R2, where V(x) is a 1-periodic function, Iμ = 1 |x| μ , μ ∈ (0, 2) and F is the primitive of f . The main feature of this paper is the nonlinearity f satisfies the critical exponential growth with respect to Trudinger-Moser inequality. Under some appropriate assumptions on g and f , by using a change of variables, a version of TrudingerMoser inequality and Hardy-Littlewood-Sobolev inequality, we obtain the existence of least energy solutions with subcritical exponential growth and Nehari-type ground state solutions with critical exponential growth via monotonicity condition instead of Ambrosetti-Rabinowitz condition. In particular, we introduce a general lower bound of t F(t) eζ0t2α near infinity, and our results extend some ones discussed in Chen et al.

  [J. Geom. Anal. 33 (2023) 299].