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

TL;DR

This work studies a nonlocal Schrödinger–Poisson–Slater type PDE $-\Delta u + (I_\alpha{\star}|u|^p)|u|^{p-2}u = f(|x|,u)$ in $\mathbb{R}^N$ within the Coulomb–Sobolev space $E^{\alpha,p}_{rad}$, leveraging a novel scaling-based variational approach. The authors formulate a nonlinear, scale-invariant eigenvalue problem and define an unbounded sequence of eigenvalues $\{\lambda_k\}$ via the $\mathbb{Z}_2$-cohomological index of Fadell and Rabinowitz, enabling new critical-point estimates and scaling-based linking arguments. They classify the local nonlinearity $f$ into subscaled, asymptotically scaled, and superscaled regimes (with a Sobolev-critical regime as well) and establish compactness of the associated energy functionals, proving existence and multiplicity results that depend on resonance between $f$ and the scaled eigenvalues. The results cover Sobolev-subcritical, critical, and supercritical growth, extending prior work by incorporating general nonlinearities and a broad parameter range, thereby providing a versatile framework for nonlocal variational problems in Coulomb–Sobolev spaces.

Abstract

We prove existence and multiplicity results for the nonlinear and nonlocal PDE $$ - Δu + (I_α\star |u|^p)\, |u|^{p-2}\, u = f(|x|,u) \quad \textrm{in} \,\,\mathbb {R}^N, $$ where $N \geq 2$, $I_α: \mathbb{R}^N \setminus \{0\} \rightarrow \mathbb{R}$ is the Riesz potential of order $α\in (1,N),$ $p>1,$ and the local nonlinearity $f: [0,\infty) \times \mathbb{R} \rightarrow \mathbb R$ is subject to a new class of assumptions. We find solutions to this zero-mass problem in a Coulomb-Sobolev space using a new scaling based approach in critical point theory, by which we classify the possibly different behaviour of the nonlinearity $f$ at zero and at infinity in terms of the scaling properties of the left hand side of the equation. This is accomplished identifying a scaling invariant PDE which can be interpreted as a nonlinear eigenvalue problem, for which a sequence of eigenvalues $\{λ_k\}$ is conveniently defined via the ${\mathbb{Z}}_2$-cohomological index of Fadell and Rabinowitz. This index allows us to use new critical group estimates (and scaling-based linking sets) which might not be possible via the classical genus. Within a fairly broad set of parameters $N,α, p$ and class of assumptions on the local nonlinearity $f,$ we establish compactness results for an associated action functional and find multiple solutions as critical points, whose existence and number is sensitive to the ''resonance'' of $f$ with the sequence of eigenvalues for the scaling invariant problem, a construction which is at places reminiscent, in the present nonlinear setting, of the classical Fredholm alternative. As a byproduct of our analysis, letting $p\neq 2$ allows us to capture general nonlinearities $f$ of Sobolev-subcritical, critical, or supercritical growth.

Theorems & Definitions (65)

• Theorem 2.1: MR3568051
• Remark 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7
• Corollary 2.8
• Theorem 2.9
• Theorem 2.10