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On the $N$-dimensional Schrödinger--Poisson--Slater equation à la Brezis--Nirenberg

Kanishka Perera, Kaye Silva

TL;DR

This work addresses the existence and multiplicity of solutions to the $N$-dimensional Schrödinger--Poisson--Slater equation with a nonlocal Hartree term and critical nonlinearity under prescribed energy. It develops a variational framework on radial Coulomb--Sobolev spaces and introduces a scaled Nehari manifold as a natural constraint for prescribing energy $c$, aided by Pohozaev's identity. A min-max scheme based on cohomological index yields a sequence of energy levels $\lambda_{c,k}$ that drive the existence of at least $k$ solutions for suitable parameter ranges, with detailed analysis at and above the critical energy $c^*$. The paper also employs Talenti-type bubble estimates to characterize the behavior of the constrained energy as $c$ approaches $0$ and $c^*$, extending Brezis--Nirenberg-type techniques to a nonlocal, critical setting. Overall, it provides a robust prescribed-energy multiplicity theory for nonlocal critical problems.

Abstract

With aid of the Pohozaev's identity and Nehari manifold, we prove the existence and multiplicity of solutions to $N$-dimensional Schrödinger--Poisson--Slater type equations involving critical exponents, by considering prescribed energy solutions.

On the $N$-dimensional Schrödinger--Poisson--Slater equation à la Brezis--Nirenberg

TL;DR

This work addresses the existence and multiplicity of solutions to the -dimensional Schrödinger--Poisson--Slater equation with a nonlocal Hartree term and critical nonlinearity under prescribed energy. It develops a variational framework on radial Coulomb--Sobolev spaces and introduces a scaled Nehari manifold as a natural constraint for prescribing energy , aided by Pohozaev's identity. A min-max scheme based on cohomological index yields a sequence of energy levels that drive the existence of at least solutions for suitable parameter ranges, with detailed analysis at and above the critical energy . The paper also employs Talenti-type bubble estimates to characterize the behavior of the constrained energy as approaches and , extending Brezis--Nirenberg-type techniques to a nonlocal, critical setting. Overall, it provides a robust prescribed-energy multiplicity theory for nonlocal critical problems.

Abstract

With aid of the Pohozaev's identity and Nehari manifold, we prove the existence and multiplicity of solutions to -dimensional Schrödinger--Poisson--Slater type equations involving critical exponents, by considering prescribed energy solutions.
Paper Structure (6 sections, 28 theorems, 92 equations, 2 figures)

This paper contains 6 sections, 28 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Technical Results
  3. Topological Properties of $\lambda_c$
  4. Estimates on Talenti Functions and Study of $\Lambda_{c}$, $c\ge c^*$
  5. Analysis of the curves $\lambda_{c,k}$
  6. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Suppose $r\in [q,2^*)$ and Then, there exists a sequence $\tilde{\lambda}_k$ satisfying $\tilde{\lambda}_k\ge 0$, $\tilde{\lambda}_k\le \tilde{\lambda}_{k+1}$ for all $k\in \mathbb{N}$, and such that:

Figures (2)

  • Figure 1: Energy curves Theorem \ref{['thm2']}
  • Figure 2: Energy curves Theorem \ref{['thm3']}

Theorems & Definitions (54)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 44 more