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Multiplicity Results for Mixed Local Nonlocal Equations With Indefinite Concave-Convex Type Nonlinearity

R. Dhanya, Jacques Giacomoni, Ritabrata Jana

TL;DR

The paper addresses multiplicity and nonexistence for a mixed local–nonlocal quasilinear equation with indefinite concave–convex nonlinearities. It develops a robust variational framework based on the Nehari manifold and fibering maps, leveraging constrained minimization and the Palais–Smale condition to obtain two distinct nonnegative weak solutions for small parameters in both subcritical and critical regimes, and it identifies threshold phenomena that preclude nontrivial solutions for large parameters via a generalized eigenvalue analysis. The results extend the theory of mixed local–nonlocal operators with sign-changing weights, providing sharp multiplicity results in subcritical, critical, and Brezis–Nirenberg-type settings, and offering new tools for handling nonhomogeneous operators. Overall, the work advances understanding of how the interplay between local and nonlocal terms and indefinite nonlinearities shapes the solution landscape, with implications for nonlinear elliptic problems at multiple scales.

Abstract

In this article we examine the multiplicity of non-negative solutions to mixed local-nonlocal equations involving \((-Δ_p) + (-Δ^{s}_{q})\) in a bounded smooth domain. The nonlinearity incorporates a parameter \(λ> 0\), a sublinear term, and a superlinear term, with sign-changing weight functions \(a(x)\) and \(b(x)\). Under suitable conditions, we establish the existence of at least two distinct nontrivial non-negative solutions in both the subcritical and critical regimes via fibering map analysis and constrained minimization on the Nehari manifold. Additionally, for \(p \not = q\), we obtain a nonexistence result for large \(λ\) by analyzing the associated generalized eigenvalue problem.

Multiplicity Results for Mixed Local Nonlocal Equations With Indefinite Concave-Convex Type Nonlinearity

TL;DR

The paper addresses multiplicity and nonexistence for a mixed local–nonlocal quasilinear equation with indefinite concave–convex nonlinearities. It develops a robust variational framework based on the Nehari manifold and fibering maps, leveraging constrained minimization and the Palais–Smale condition to obtain two distinct nonnegative weak solutions for small parameters in both subcritical and critical regimes, and it identifies threshold phenomena that preclude nontrivial solutions for large parameters via a generalized eigenvalue analysis. The results extend the theory of mixed local–nonlocal operators with sign-changing weights, providing sharp multiplicity results in subcritical, critical, and Brezis–Nirenberg-type settings, and offering new tools for handling nonhomogeneous operators. Overall, the work advances understanding of how the interplay between local and nonlocal terms and indefinite nonlinearities shapes the solution landscape, with implications for nonlinear elliptic problems at multiple scales.

Abstract

In this article we examine the multiplicity of non-negative solutions to mixed local-nonlocal equations involving \((-Δ_p) + (-Δ^{s}_{q})\) in a bounded smooth domain. The nonlinearity incorporates a parameter , a sublinear term, and a superlinear term, with sign-changing weight functions \(a(x)\) and \(b(x)\). Under suitable conditions, we establish the existence of at least two distinct nontrivial non-negative solutions in both the subcritical and critical regimes via fibering map analysis and constrained minimization on the Nehari manifold. Additionally, for , we obtain a nonexistence result for large by analyzing the associated generalized eigenvalue problem.

Paper Structure

This paper contains 12 sections, 34 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Nehari Manifold and Fibering Map Analysis
  4. The Nehari Manifold
  5. Fibering Map Analysis
  6. The Palais–Smale condition
  7. Multiplicity Result When $p\leq q$ and $r<\max\{{p_{*}},q_{s}^{*}\}$
  8. Nonexistence Result
  9. Nonexistence Result For Generalized Eigenvalue Problem
  10. Nonexistence result
  11. Multiplicity Result When $q\leq p$ and $r=p_{*}$
  12. Brezis Nirenberg Type Problem

Key Result

Theorem 1.1

Let $p \leq q$ and $r < \max\{{p_{*}},q_{s}^{*}\}$. Additionally, suppose that $a(x)\in L^{\frac{r}{r-\delta}}(\Omega)$ and $b(x)\in L^{\infty}(\Omega)$. Then, there exists $\lambda_0 > 0$ such that for all $\lambda \in (0, \lambda_0)$, the problem generaleqn admits at least two nonnegative, non-tri

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • ...and 46 more