Schrodinger-Poisson-Slater equations with nonlinearity subscaled near zero

TL;DR

This paper investigates solutions to a Schrödinger-Poisson-Slater equation with a nonlinear term that behaves like a power near zero, using Morse theory and critical point analysis to establish existence results.

Contribution

It introduces new critical point results for the Schrödinger-Poisson-Slater equation with subscaled nonlinearity near zero, applying Morse theory and abstract critical group analysis.

Findings

Existence of nonzero solutions via Morse theory.

Sequence of solutions for odd nonlinearities using Clark's theorem.

Development of an abstract critical groups at infinity result.

Abstract

We study the following zero-mass Schr{\"o}dinger-Poisson-Slater equation \[ - \Delta u + \left( \frac{1}{4 \pi | x |} \ast u^2 \right) u = f (| x |, u) \text{,} \qquad u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] with nonlinearity subscaled near zero in the sense that $f (| x |, t) \approx a | t |^{p - 2} t$ as $| t | \rightarrow 0$ for some $p\in\big(\frac{18}{7},3\big)$. A nonzero solution is obtained via Morse theory when the nonlinearity is asymptotically scaled at infinity. For this purpose we prove an abstract result on the critical groups at infinity for functionals satisfying the geometric assumptions of the scaled saddle point theorem of Mercuri \& Perera [arXiv:2411.15887]. For the case that $f (| x |, \cdot)$ is odd, a sequence of solutions are obtained via a version of Clark's theorem due to Kajikiya [J.\ Funct.\ Anal.\ 225 (2005) 352--370].