Sharp Trudinger–Moser Inequality and Ground State Solutions to Quasi-Linear Schrödinger Equations with Degenerate Potentials in ℝ
n
Abstract The main purpose of this paper is to establish the existence of ground-state solutions to a class of Schrödinger equations with critical exponential growth involving the nonnegative, possibly degenerate, potential V: - div ( | ∇ u | n - 2 ∇ u ) + V ( x ) | u | n - 2 u = f ( u ) . -\operatorname{div}(\lvert\nabla u\rvert^{n-2}\nabla u)+V(x)\lvert u\rvert^{n-% 2}u=f(u). To this end, we first need to prove a sharp Trudinger–Moser inequality in ℝ n {\mathbb{R}^{n}} under the constraint ∫ ℝ n ( | ∇ u | n + V ( x ) | u | n ) 𝑑 x ≤ 1 . \int_{\mathbb{R}^{n}}(\lvert\nabla u\rvert^{n}+V(x)\lvert u\rvert^{n})\,dx\leq 1. This is proved without using the technique of blow-up analysis or symmetrization argument. As far as what has been studied in the literature, having a positive lower bound has become a standard assumption on the potential V ( x ) {V(x)} in dealing with the existence of solutions to the above Schrödinger equation. Since V ( x ) {V(x)} is allowed to vanish on an open set in ℝ n {\mathbb{R}^{n}} , the loss of a positive lower bound of the potential V ( x ) {V(x)} makes this problem become fairly nontrivial. Our method to prove the Trudinger–Moser inequality in ℝ 2 {\mathbb{R}^{2}} (see [L. Chen, G. Lu and M. Zhu, A critical Trudinger–Moser inequality involving a degenerate potential and nonlinear Schrödinger equations, Sci. China Math. 64 2021, 7, 1391–1410]) does not apply to this higher-dimensional case ℝ n {\mathbb{R}^{n}} for n ≥ 3 {n\geq 3} here. To obtain the existence of a ground state solution, we use a non-symmetric argument to exclude the possibilities of vanishing and dichotomy cases of the minimizing sequence in the Nehari manifold. This argument is much simpler than the one used in dimension two where we consider the nonlinear Schrödinger equation - Δ u + V u = f ( u ) {-\Delta u+Vu=f(u)} with a degenerate potential V in ℝ 2 {\mathbb{R}^{2}} .
