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Multiplicity of positive solutions for mixed local-nonlocal singular critical problems

Stefano Biagi, Eugenio Vecchi

Abstract

We prove the existence of at least two positive weak solutions for mixed local-nonlocal singular and critical semilinear elliptic problems in the spirit of [Haitao, 2003], extending the recent results in [Garain, 2023] concerning singular problems and, at the same time, the results in [Biagi, Dipierro, Valdinoci, Vecchi, 2022] regarding critical problems.

Multiplicity of positive solutions for mixed local-nonlocal singular critical problems

Abstract

We prove the existence of at least two positive weak solutions for mixed local-nonlocal singular and critical semilinear elliptic problems in the spirit of [Haitao, 2003], extending the recent results in [Garain, 2023] concerning singular problems and, at the same time, the results in [Biagi, Dipierro, Valdinoci, Vecchi, 2022] regarding critical problems.
Paper Structure (5 sections, 17 theorems, 217 equations)

This paper contains 5 sections, 17 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some auxiliary results
  4. Proof of Theorem \ref{['thm:main']}
  5. Existence of a second solution

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^n$(with $n\geq 3$) be a bounded open set with smooth enough boundary, and let $\gamma\in (0,1)$. Then, for every $\varepsilon \in (0,1]$, there exists $\Lambda_{\varepsilon} > 0$ such that Moreover, for every $\lambda \in (0,\Lambda_{\varepsilon})$, the weak solution $u_{\lambda}$ found above is a local minimizer of the functional in the $\mathcal{X}^{1,2}(\Omega)$-t

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 2.1: Properties of the space $\mathcal{X}^{1,2}(\Omega)$
  • Definition 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • Lemma 3.4
  • ...and 29 more