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On the existence of a second positive solution to mixed local-nonlocal concave-convex critical problems

Stefano Biagi, Eugenio Vecchi

Abstract

We prove the existence of a second positive weak solution for mixed local-nonlocal critical semilinear elliptic problems with a sublinear perturbation in the spirit of [Ambrosetti, Brezis, Cerami, 1994].

On the existence of a second positive solution to mixed local-nonlocal concave-convex critical problems

Abstract

We prove the existence of a second positive weak solution for mixed local-nonlocal critical semilinear elliptic problems with a sublinear perturbation in the spirit of [Ambrosetti, Brezis, Cerami, 1994].
Paper Structure (2 sections, 6 theorems, 25 equations)

This paper contains 2 sections, 6 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries

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 $p \in (0,1)$. Then, there exists $\Lambda > 0$ such that Moreover, for $\lambda \in (0,\Lambda)$, the solution is minimal and increasing w.r.t. to $\lambda$.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Remark 2.1: Properties of the space $\mathcal{X}^{1,2}(\Omega)$
  • Definition 2.2