Nonlinear scalar field equations with a critical Hardy potential

TL;DR

This paper proves the existence of solutions for a nonlinear scalar field equation with a critical Hardy potential using variational methods, including non-radial solutions under symmetry assumptions.

Contribution

It introduces a variational approach in a new functional setting to establish solutions for the critical Hardy potential problem, including non-radial solutions.

Findings

Existence of a nontrivial solution minimizing the energy on the Pohožaev manifold.

Solutions are found in a specialized functional space tailored to the critical potential.

Under symmetry assumptions, at least one non-radial solution exists.

Abstract

We study the existence of solutions for the nonlinear scalar field equation $$-\Delta u - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2)^2}{4|x|^2}$ is the critical Hardy potential and $N \geq 3$. The nonlinearity $g$ is continuous and satisfies general subcritical growth assumptions of the Berestycki-Lions type. The problem is approached using variational methods within a non-standard functional setting. The natural energy functional associated with the equation is defined on the space $X^1(\mathbb{R}^N)$, which is the completion of $H^1(\mathbb{R}^N)$ with respect to the norm induced by the quadratic part of the functional. We establish the existence of a nontrivial solution $u_0 \in X^1(\mathbb{R}^N)$ that satisfies the Poho\v{z}aev constraint $\mathcal{M}$ and minimizes the energy functional on $\mathcal{M}$. Furthermore, assuming $g$ is odd, we prove the existence of at least one non-radial solution.