New solutions to Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces

Technical Summary: New Solutions to Schrödinger–Poisson–Slater Equations in Coulomb-Sobolev Spaces

Problem Statement
The paper investigates the existence and multiplicity of solutions to the nonlinear and nonlocal partial differential equation:
$$-\Delta u + (I_\alpha \star |u|^p) |u|^{p-2} u = f(|x|, u) \quad \text{in } \mathbb{R}^N,$$
where $N \ge 2$, $p > 1$, and $\alpha \in (1, N)$. Here, $I_\alpha$ denotes the Riesz potential of order $\alpha$. The nonlinearity $f: [0, \infty) \times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function subject to a new, broad class of assumptions.

This equation models a zero-mass problem arising in quantum many-body systems (Hartree–Fock approximations) and semiconductor models. Unlike positive-mass variants, this problem is naturally set in a Coulomb-Sobolev space $E_{\alpha, p}(\mathbb{R}^N)$, defined by the norm:
$$\|u\| = \left[ \int_{\mathbb{R}^N} |\nabla u|^2 dx + \left( \int_{\mathbb{R}^N} |I_{\alpha/2} \star |u|^p|^2 dx \right)^{1/p} \right]^{1/2}.$$
The authors work within the subspace of radial functions, $E_{\alpha, p}^{\text{rad}}(\mathbb{R}^N)$, to recover compactness properties necessary for variational methods. The study addresses the loss of compactness in embeddings into Lebesgue spaces, governed by critical exponents such as the Coulomb-Sobolev exponent $q_{CS} = 2(2p+\alpha)/(2+\alpha)$ and the radial exponent $q_{rad}$.

Methodology
The primary methodological innovation is a scaling-based approach within critical point theory. The authors adapt techniques from Mercuri and Perera [28] to handle situations where classical variational geometries (such as the mountain-pass geometry or linking based on linear subspaces) may not be present or where $Z_2$ symmetry is insufficient.

Key methodological components include:

1. Scaling Invariance: The authors identify a scaling invariant nonlinear eigenvalue problem:
   $$-\Delta u + (I_\alpha \star |u|^p) |u|^{p-2} u = \lambda |u|^{q-2} u,$$
   where $q = q_{CS}$. This problem admits a sequence of eigenvalues $\{\lambda_k\}$ defined via the $Z_2$-cohomological index of Fadell and Rabinowitz, rather than the classical genus.
2. Classification of Nonlinearities: The behavior of the local nonlinearity $f$ is classified relative to the scaling properties of the left-hand side of the equation. Depending on the limit of $f(|x|, t)/|t|^{q-2}t$ as $|t| \to \infty$ (or $0$),$f$ is categorized as:
   • Subscaled: The nonlinearity is dominated by the principal part at infinity (or zero).
   • Asymptotically Scaled: The nonlinearity behaves like an eigenvalue term at infinity (or zero).
   • Superscaled: The nonlinearity dominates the principal part.
3. Critical Group Estimates: By utilizing the cohomological index, the authors derive new critical group estimates and construct scaling-based linking sets. This allows for the detection of critical points even when the functional lacks the standard symmetries required for classical index theories.
4. Compactness Analysis: The paper establishes compactness results (Palais-Smale conditions) for the associated action functional across various parameter regimes ($N, \alpha, p$), addressing the challenges posed by Sobolev-subcritical, critical, and supercritical growth.

Key Results
The paper establishes existence and multiplicity theorems for the zero-mass problem under various growth conditions on $f$:

• Existence in Subscaled Regimes: For nonlinearities that are "subscaled" (dominated by the principal part), the authors prove the existence of at least one solution.
• Multiplicity via Resonance: When $f$ is asymptotically scaled (approaching an eigenvalue term $\lambda |u|^{q-2}u$), the existence of nontrivial solutions depends on the position of $\lambda$ relative to the eigenvalues $\{\lambda_k\}$.
  • If $\lambda > \lambda_1$, a nontrivial solution exists.
  • If $\lambda > \lambda_2$, two nontrivial solutions exist.
  • These results hold even if the primitive of the perturbation term is negative, removing the need for $\lambda$ to be non-resonant in certain cases.
• Superscaled and Critical Cases:
  • In the superscaled case, where the nonlinearity dominates, the authors prove the existence of nontrivial solutions under sign conditions on the primitive of the perturbation, or by introducing a small parameter.
  • In the critical case (involving the critical Sobolev exponent $2^*$), the paper proves the existence of $m$ distinct pairs of solutions for parameters near eigenvalues with multiplicity $m$.
  • The number of solutions can be made arbitrarily large by adjusting parameters (e.g., the coefficient of a subcritical term), with the solution count tending to infinity as the parameter grows.

Significance and Claims
The authors claim that their work extends recent results by Mercuri and Perera [28] to a broader range of parameters $N, \alpha, p$ and a more general class of nonlinearities $f$. The significance of the paper lies in:

1. Generalization of Variational Regimes: The introduction of the terms "subscaled," "scaled," and "superscaled" provides a unified framework to classify variational regimes that were previously treated separately or not at all. This classification reflects the interaction between the nonlinearity $f$ and the scaling properties of the equation.
2. Handling Non-Symmetric and Non-Standard Geometries: The approach allows for the treatment of problems where the energy functional does not exhibit standard mountain-pass geometry or where the nonlinearity is not an odd function (preventing the use of standard symmetric critical point theorems).
3. Capturing Supercritical Growth: By working in the Coulomb-Sobolev space with $p \neq 2$, the methodology captures nonlinearities of Sobolev-subcritical, critical, and supercritical growth, which are often difficult to handle in classical Sobolev settings.
4. Analogy to Fredholm Alternative: The classification of solutions based on the "resonance" of $f$ with the sequence of eigenvalues $\{\lambda_k\}$ is described as a nonlinear analogue of the classical Fredholm alternative.

The paper does not propose new physical applications or experimental proposals but rather provides a rigorous mathematical foundation for solving these equations under extended physical parameters found in Density Functional Theory (DFT) and quantum chemistry models. The authors note that while they use the assumption $2p < 2^*$ in some results to ensure boundedness of Palais-Smale sequences, they hope to relax this condition in future work.