An Existence Result for a Generalized Quasilinear Schrödinger Equation with Nonlocal Term. (6th October 2020) - Li, Quanqing; Teng, Kaimin; Zhang, Jian; Nie, Jianjun
Abstract : In this paper, we consider the following generalized quasilinear Schrödinger equation with nonlocal term − div g 2 u ∇ u + g u g ′ u ∇ u 2 + V x u = λ x − μ ∗ u p u p − 2 u, x ∈ ℝ N, where N ≥ 3, g : ℝ → ℝ + is a C 1 even function, g 0 = 1, g ′ s ≥ 0 is for all s ≥ 0, lim ∣ s ∣ → + ∞ g s / s α − 1 ≔ β > 0 is for some α > 1, and α − 1 g s ≥ g ′ s s is for all s ≥ 0, 2 α ≤ p ≤ 2 α N − μ / N − 2, and 0 < μ < N . We prove that the equation admits a solution by using a constrained minimization argument.