Nonlinear Schrödinger equations with a critical, inverse-square potential

TL;DR

The paper addresses the existence of standing-wave solutions to a nonlinear Schrödinger equation with a critical inverse-square (Hardy) potential in $\mathbb{R}^N$ for $N\ge 3$, with $\mathbb{Z}^N$-periodic $V$ and subcritical, superlinear nonlinearity $f$. It develops a variational framework in the nontranslation-invariant space $X^1(\mathbb{R}^N)$ and introduces a novel profile decomposition adapted to the Hardy term to recover compactness. A Nehari-manifold-based Cerami framework yields a bounded Cerami sequence at a positive level, and the new profile decomposition shows that any loss of compactness can be ruled out via energy splitting, forcing strong convergence to a ground state. The result extends prior subcritical analyses to the critical Hardy regime, demonstrating the existence of a ground-state solution in a challenging noncompact setting with periodic coefficients.

Abstract

We study the existence of solutions of the following nonlinear Schrödinger equation $$ -Δu+V(x)u-\frac{(N-2)^2}{4|x|^2}u=f(x,u) $$ where $V:\mathbb{R}^N\to\mathbb{R}$ and $f:\mathbb{R}^N\times \mathbb{R}\to \mathbb{R}$ are periodic with respect to $x\in\mathbb{R}^N.$ We assume that $V$ has positive essential infimum, $f$ satisfies weak growth conditions and $N\geq 3$. The approach to the problem uses variational methods with nonstandard functional setting. We obtain the existence of the ground state solution using the new profile decomposition.

Theorems & Definitions (17)

• Theorem 1.1
• Lemma 2.1
• Theorem 3.1: BB
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 4.1: BBDS