Sharp Trudinger–Moser Inequality and Ground State Solutions to Quasi-Linear Schrödinger Equations with Degenerate Potentials in ℝ n

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 ) . -div(∇ uⁿ⁻²∇ u)+V(x) un-% 2u=f(u). To this end, we first need to prove a sharp Trudinger–Moser inequality in ℝ n {Rⁿ} under the constraint ∫ ℝ n ( | ∇ ⁡ u | n + V ⁢ ( x ) ⁢ | u | n ) ⁢ 𝑑 x ≤ 1 .