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Multiplicity of solutions for singular elliptic problems with Stein-Weiss term

Márcia S. B. A. Cardoso, Edcarlos D. Silva, Marcos. L. M. Carvalho, Minbo Yang

TL;DR

The paper investigates a singular elliptic problem on $\mathbb{R}^N$ with a double weighted Stein-Weiss nonlocal term, proving the existence and multiplicity of positive weak solutions despite the energy being non-differentiable. By combining the Nehari manifold approach with a generalized nonlinear Rayleigh quotient, the authors identify an extremal parameter $\lambda^*>0$ and show that for all $\lambda\in(0,\lambda^*)$ there are at least two distinct positive solutions, one a ground state and the other a bound state; at $\lambda=\lambda^*$, two positive solutions persist (one in each Nehari component) while ruling out degenerate inflection points. The analysis hinges on Stein-Weiss and weighted Hardy-Littlewood-Sobolev inequalities, careful variational estimates, and a limiting argument as $\lambda\uparrow\lambda^*$. The results extend the multiplicity theory to problems with a Stein-Weiss term in the whole space and singular nonlinearities, providing a framework for extremal parameter selection and continuity of solution branches. This advances understanding of nonlocal, singular elliptic equations and their critical point structure in unbounded domains.

Abstract

In the present work, we establish the existence and multiplicity of positive solutions for the singular elliptic equations with a double weighted nonlocal interaction term defined in the whole space $\mathbb{R}^N$. The nonlocal term and the fact that the energy functional is not differentiable are the main difficulties for this kind of problem. We apply the Nehari method and the nonlinear Rayleigh quotient to prove that our main problem has at least two positive weak solutions. Furthermore, we prove a nonexistence result related to the extreme $λ^*> 0$ given by the nonlinear Rayleigh quotient.

Multiplicity of solutions for singular elliptic problems with Stein-Weiss term

TL;DR

The paper investigates a singular elliptic problem on with a double weighted Stein-Weiss nonlocal term, proving the existence and multiplicity of positive weak solutions despite the energy being non-differentiable. By combining the Nehari manifold approach with a generalized nonlinear Rayleigh quotient, the authors identify an extremal parameter and show that for all there are at least two distinct positive solutions, one a ground state and the other a bound state; at , two positive solutions persist (one in each Nehari component) while ruling out degenerate inflection points. The analysis hinges on Stein-Weiss and weighted Hardy-Littlewood-Sobolev inequalities, careful variational estimates, and a limiting argument as . The results extend the multiplicity theory to problems with a Stein-Weiss term in the whole space and singular nonlinearities, providing a framework for extremal parameter selection and continuity of solution branches. This advances understanding of nonlocal, singular elliptic equations and their critical point structure in unbounded domains.

Abstract

In the present work, we establish the existence and multiplicity of positive solutions for the singular elliptic equations with a double weighted nonlocal interaction term defined in the whole space . The nonlocal term and the fact that the energy functional is not differentiable are the main difficulties for this kind of problem. We apply the Nehari method and the nonlinear Rayleigh quotient to prove that our main problem has at least two positive weak solutions. Furthermore, we prove a nonexistence result related to the extreme given by the nonlinear Rayleigh quotient.

Paper Structure

This paper contains 10 sections, 27 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main theorems
  3. Notations
  4. Outline
  5. Preliminaries
  6. Generalized nonlinear Rayleigh quotient method
  7. Proof of Theorem \ref{['teorema 1']}
  8. Multiplicity of solutions for $\lambda=\lambda^*$
  9. The proof of Theorem \ref{['teorema 2']} completed
  10. Declarations

Key Result

Theorem 1.1

Suppose ($H_0$), ($V_1$), ($V_2$). Then $0<\lambda_*<\lambda^*<\infty$ and for each $\lambda\in (0,\lambda^*)$ the Problem P1 admits at least two distinct positive solutions $u_{\lambda},w_\lambda \in H^1(\mathbb R^N)$. Furthermore, we obtain the following statements:

Theorems & Definitions (51)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1: Stein-Weiss Inequality
  • Proposition 2.2: Weighted Hardy-Littlewood-Sobolev Inequality
  • Lemma 2.1
  • Lemma 2.2
  • ...and 41 more