Schrodinger-Poisson-Slater equations with nonlinearity subscaled near zero

TL;DR

The paper addresses the zero-mass Schrödinger–Poisson–Slater equation in $\mathbb{R}^3$ with a subscaled near-zero nonlinearity and develops a variational framework based on $3$-scaling to obtain a nonzero solution via scaled critical-group theory, plus a Clark/Kajikiya-type sequence of small-energy solutions for odd nonlinearities, and an extension to mixed-power nonlinearities through a new radial space $X$. The key contributions include an abstract result on critical groups at infinity and a systematic use of scaled geometry to circumvent standard Palais–Smale obstacles. The results broaden existence and multiplicity results for nonlocal nonlinear Schrödinger equations under near-zero subcritical growth and scaling assumptions, with potential implications for Hartree–Fock-type models.

Abstract

We study the following zero-mass Schr{ö}dinger-Poisson-Slater equation \[ - Δu + \left( \frac{1}{4 π| x |} \ast u^2 \right) u = f (| x |, u) \text{,} \qquad u \in \mathcal{D}^{1, 2} (\mathbb{R}^3) \text{} \] 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].

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.4
• Proposition 2.1: MR1420790
• Theorem 2.2
• Lemma 3.1
• Proposition 4.1: MR2152503
• Lemma 4.3
• Lemma 5.1
• Lemma 5.2