New solutions to Schr\"{o}dinger-Poisson-Slater equations in Coulomb-Sobolev spaces

TL;DR

This paper establishes existence and multiplicity of solutions for a class of nonlinear, nonlocal Schrödinger-Poisson-Slater equations in Coulomb-Sobolev spaces, using a novel scaling approach and topological index theory.

Contribution

It introduces a new scaling-based critical point framework and applies cohomological index theory to analyze nonlinear eigenvalues for these equations.

Findings

Proves existence of multiple solutions under broad parameter conditions.

Develops a new classification of nonlinearities based on scaling properties.

Identifies a sequence of eigenvalues influencing solution multiplicity.

Abstract

We prove existence and multiplicity results for the nonlinear and nonlocal PDE $$ - \Delta u + (I_\alpha \star |u|^p)\, |u|^{p-2}\, u = f(|x|,u) \quad \textrm{in} \,\,\mathbb {R}^N, $$ where $N \geq 2$, $I_\alpha : \mathbb{R}^N \setminus \{0\} \rightarrow \mathbb{R}$ is the Riesz potential of order $\alpha \in (1,N),$ $p>1,$ 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 $\{\lambda_k\}$ 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 $N,\alpha, p$ 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.