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Prescribed energy solutions to some scaled problems via a scaled Nehari manifold

Kanishka Perera, Kaye Silva

TL;DR

The paper introduces a scaling-based Nehari framework to construct solutions with prescribed energy for a broad class of nonlocal nonlinear elliptic problems, including Schrödinger–Poisson–Slater type equations. By developing a scaled eigenvalue theory, a scaled Nehari manifold, and associated minimax principles, it establishes existence, multiplicity, and bifurcation results across subscaled, superscaled, and mixed nonlinearities. The authors define energy-curves $C_k$ and show how intersections with the energy line yield energy-constrained critical points, providing a unified approach applicable to density functional theory models and related nonlocal problems. The work also verifies the abstract hypotheses for the concrete Schrödinger–Poisson–Slater setting via Pohožaev identities and culminates in Theorems 101–106, with potential extensions to other operators and fractional settings.

Abstract

We prove the existence, multiplicity, and bifurcation of solutions with prescribed energy for a broad class of scaled problems by introducing a suitable notion of scaling based Nehari manifold. Applications are given to Schrödinger--Poisson--Slater type equations.

Prescribed energy solutions to some scaled problems via a scaled Nehari manifold

TL;DR

The paper introduces a scaling-based Nehari framework to construct solutions with prescribed energy for a broad class of nonlocal nonlinear elliptic problems, including Schrödinger–Poisson–Slater type equations. By developing a scaled eigenvalue theory, a scaled Nehari manifold, and associated minimax principles, it establishes existence, multiplicity, and bifurcation results across subscaled, superscaled, and mixed nonlinearities. The authors define energy-curves and show how intersections with the energy line yield energy-constrained critical points, providing a unified approach applicable to density functional theory models and related nonlocal problems. The work also verifies the abstract hypotheses for the concrete Schrödinger–Poisson–Slater setting via Pohožaev identities and culminates in Theorems 101–106, with potential extensions to other operators and fractional settings.

Abstract

We prove the existence, multiplicity, and bifurcation of solutions with prescribed energy for a broad class of scaled problems by introducing a suitable notion of scaling based Nehari manifold. Applications are given to Schrödinger--Poisson--Slater type equations.

Paper Structure

This paper contains 9 sections, 28 theorems, 130 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Abstract results
  3. Scaled eigenvalue problems
  4. Formulation of the problem
  5. Scaled Nehari manifold
  6. Minimax critical values
  7. Analysis of the curves
  8. Main results
  9. Proofs of Theorems \ref{['Theorem 101']}--\ref{['Theorem 106']}

Key Result

Theorem 1.1

Let $18/7 < \sigma < 3$.

Figures (6)

  • Figure 1: Energy curves for the case $F>0,\, G=0$ (see Theorem \ref{['Theorem 1']}).
  • Figure 2: Energy curves for the case $F<0,\, G=0$ (see Theorem \ref{['Theorem 2']}).
  • Figure 3: Energy curves for the case $F=0,\, G>0$ (see Theorem \ref{['Theorem 3']}).
  • Figure 4: Energy curves for the case $F=0,\, G<0$ (see Theorem \ref{['Theorem 4']}).
  • Figure 5: Energy curves for the case $F>0,\, G<0$ (see Theorem \ref{['Theorem 5']}).
  • ...and 1 more figures

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1: MePe2
  • Definition 2.2: MePe2
  • Theorem 2.3: MePe2
  • Remark 2.4
  • ...and 41 more