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

Abstract

We prove existence and multiplicity results for the nonlinear and nonlocal PDE $$-\Delta u+(I_{\alpha }\star |u{|}^p)|u{|}^{p-2}u=f(|x|,u)\text{in}{\mathbb{R}}^N,$$ where $N\ge 2$, $I_{\alpha }:{\mathbb{R}}^N\setminus \{0\}\to \mathbb{R}$ is the Riesz potential of order $\alpha \in (1,N),$ $p>1,$ and the local nonlinearity $f:[0,\infty )\times \mathbb{R}\to \mathbb{R}$ 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 $p\ne 2$ allows us to capture general nonlinearities $f$ of Sobolev-subcritical, critical, or supercritical growth.

37 pages