Nonlinear scalar field equations with a critical Hardy potential

Abstract

We study the existence of solutions for the nonlinear scalar field equation $$-Δu - \frac{(N-2)^2}{4|x|^2} u = g(u), \quad \mbox{in } \mathbb{R}^N \setminus \{0\},$$ where the potential $-\frac{(N-2{)}^2}{4|x{|}^2}$ is the critical Hardy potential and $N\ge 3$. 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 Pohožaev 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.