Note on the multiplicity of solutions for nonlinear scalar field equations with a critical inverse-square potential

Technical Summary: Multiplicity of Solutions for Nonlinear Scalar Field Equations with a Critical Inverse-Square Potential

Problem Statement
The paper investigates the multiplicity of solutions to the nonlinear scalar field equation:
$$-\Delta u - \frac{(N-2)^2}{4|x|^2}u = g(u), \quad \text{in } \mathbb{R}^N \setminus \{0\},$$
where $N \geq 3$. The equation features the critical inverse-square Hardy potential with the optimal constant $\mu_c = (N-2)^2/4$. This specific coefficient places the problem at the critical threshold of the Hardy inequality, introducing significant analytical challenges. The nonlinearity $g: \mathbb{R} \to \mathbb{R}$ is assumed to be continuous, odd, and satisfy specific growth and sign conditions (denoted as (G1)–(G5)), which include subcritical growth at infinity and a superlinear behavior near zero.

Methodology and Functional Framework
The authors build upon the functional framework established in their previous work [3] and apply abstract critical point theory developed by Ikoma [11].

1. Energy Space ($X^1$): Due to the lack of coercivity of the quadratic form associated with the operator $-\Delta - \frac{(N-2)^2}{4|x|^2}$ on the standard Sobolev space $H^1(\mathbb{R}^N)$, the natural energy space is strictly larger than $H^1(\mathbb{R}^N)$. The authors utilize $X^1(\mathbb{R}^N)$, defined as the completion of $H^1(\mathbb{R}^N)$ with respect to a norm involving the Hardy term. Because the standard integrals in the energy functional are not well-defined for functions in $X^1 \setminus H^1$, the authors employ a singular bilinear form $\xi(u, v)$ involving a limiting process to define the inner product and norm on this space.
2. Symmetry and Subspaces: To establish multiplicity, the authors restrict the energy functional to specific subspaces of $X^1(\mathbb{R}^N)$:
   • Radial Case: The subspace of radially symmetric functions $X^1_{\text{rad}}(\mathbb{R}^N)$ for $N \geq 3$.
   • Non-radial Case: For $N \geq 4$ and $N \neq 5$, the authors define a subspace $X$ consisting of functions invariant under a specific group of symmetries ($O_2$) and antisymmetric with respect to an involution $\tau(x_1, x_2, x_3) = (x_2, x_1, x_3)$. This construction ensures the exclusion of non-trivial radial functions.
3. Variational Approach: The solutions are sought as critical points of the energy functional $J(u) = \frac{1}{2}\xi(u) - \int_{\mathbb{R}^N} G(u) dx$, where $G$ is the primitive of $g$. The authors utilize the Pohožaev functional $P(u)$ to define a constraint manifold. They verify the $(PPS)_c$ condition (a variant of the Palais-Smale condition incorporating the Pohožaev identity) for the restricted functional.
4. Critical Point Theorem: The existence of infinitely many solutions is derived using Ikoma's abstract critical point theorem. This theorem requires the functional to be even (guaranteed by the oddness of $g$) and satisfies specific geometric conditions (mountain pass geometry and the existence of odd maps from spheres into the function space with negative energy).

Key Contributions and Results
The primary contribution of the paper is the establishment of the existence of infinitely many distinct weak solutions to equation (1.1), categorized by their symmetry:

• Radial Solutions: For any dimension $N \geq 3$, there exists a sequence of radial weak solutions $(u_j)_{j=1}^\infty$ such that the energy $J(u_j) \to \infty$ as $j \to \infty$.
• Non-radial Solutions: For dimensions $N = 4$ or $N \geq 6$, there exists a sequence of non-radial weak solutions $(u_j)_{j=1}^\infty$ with unbounded energy.
• Regularity: The solutions obtained are shown to be of class $C^{1,\alpha}_{\text{loc}}(\mathbb{R}^N \setminus \{0\})$. If the nonlinearity $g$ is Hölder continuous, the solutions are of class $C^2(\mathbb{R}^N \setminus \{0\})$.
• Pohožaev Identity: The constructed solutions satisfy the Pohožaev identity $P(u_j) = 0$.

Significance and Claims
The paper claims to resolve the multiplicity question for the critical inverse-square potential, a regime where standard variational methods often fail due to the lack of translation invariance and the non-standard nature of the energy space.

• Extension of Previous Work: While the authors' prior work [3] established the existence of at least one non-radial solution and a radial solution (under the assumption that $g$ is odd), this paper extends those results to prove the existence of infinitely many solutions of both types.
• Overcoming Analytical Difficulties: The work demonstrates that despite the "fall to the center" phenomenon and the loss of coercivity at the critical constant, the variational structure remains robust enough to support an infinite sequence of solutions when the nonlinearity possesses odd symmetry.
• Methodological Rigor: The paper rigorously addresses the definition of the energy functional on the enlarged space $X^1$ and validates the compactness properties required for the application of abstract critical point theorems in this singular setting.

The authors explicitly note that the regularity theory in the critical Hardy setting is not fully developed, and thus it remains an open question whether all weak solutions satisfy the Pohožaev identity; however, the solutions constructed in this work are specifically selected to satisfy it.