Nonlinear scalar field equations with a critical Hardy potential
Bartosz Bieganowski, Daniel Strzelecki
TL;DR
The paper addresses the nonlinear scalar field equation with the critical Hardy potential $-\frac{(N-2)^2}{4|x|^2}$ in $\mathbb{R}^N\setminus\{0\}$, for $N\ge 3$, under Berestycki-Lions-type growth conditions on $g$. It develops a noncoercive variational framework in the enlarged space $X^1(\mathbb{R}^N)$ and uses an augmented Pohožaev-Palais-Smale approach together with a tailored profile decomposition to overcome lack of translation invariance and compactness issues, proving the existence of a nontrivial solution $u_0$ that minimizes the energy on the Pohožaev constraint ${\mathcal M}$ and satisfies the Pohožaev condition. When $g$ is odd, the authors also construct a nonradial solution by exploiting symmetry constraints, while a radial minimizer on ${\mathcal M}$ is shown to exist and be regular. Additional qualitative properties include the fact that nonnegative solutions cannot belong to $H^1(\mathbb{R}^N)$ and that the radial minimizer enjoys higher regularity. These results advance variational methods for singular, critical Hardy-type problems and establish new symmetry and regularity phenomena for such equations.
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 $N \geq 3$. 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 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.