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Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R^2

Sanjit Biswas

TL;DR

This work addresses multiplicity for a mixed local-nonlocal equation in 2D with a variable singular exponent and critical exponential nonlinearity. The authors combine sub-solution/super-solution techniques with variational methods to obtain two positive weak solutions under suitable conditions, including a threshold $\Lambda_\varepsilon$ for solvability in $\lambda$. A second main result shows the persistence of two ordered positive solutions $(u_{\lambda,\varepsilon},v_{\lambda,\varepsilon})$ for small $\varepsilon$ and $\lambda$ under stronger hypotheses, using a refined energy- and perturbation-analysis argument that leverages Moser-type test functions. The findings extend multiplicity results to mixed local-nonlocal operators with variable singularities and critical growth in dimension two, highlighting delicate energy and compactness issues in the Moser–Trudinger regime.

Abstract

In this article, we prove the existence of at least two positive weak solutions for a mixed local-nonlocal singular problem in the presence of critical exponential nonlinearity in dimension two. The novelty of this work is the inclusion of a variable singular exponent in the context of mixed operator and critical exponential nonlinearity in R^2. Our approach is based on sub-solution super-solution technique, combined with variational methods.

Multiplicity results for mixed local-nonlocal equations with singular and critical exponential nonlinearity in R^2

TL;DR

This work addresses multiplicity for a mixed local-nonlocal equation in 2D with a variable singular exponent and critical exponential nonlinearity. The authors combine sub-solution/super-solution techniques with variational methods to obtain two positive weak solutions under suitable conditions, including a threshold for solvability in . A second main result shows the persistence of two ordered positive solutions for small and under stronger hypotheses, using a refined energy- and perturbation-analysis argument that leverages Moser-type test functions. The findings extend multiplicity results to mixed local-nonlocal operators with variable singularities and critical growth in dimension two, highlighting delicate energy and compactness issues in the Moser–Trudinger regime.

Abstract

In this article, we prove the existence of at least two positive weak solutions for a mixed local-nonlocal singular problem in the presence of critical exponential nonlinearity in dimension two. The novelty of this work is the inclusion of a variable singular exponent in the context of mixed operator and critical exponential nonlinearity in R^2. Our approach is based on sub-solution super-solution technique, combined with variational methods.

Paper Structure

This paper contains 6 sections, 16 theorems, 178 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Auxiliary Results
  4. Proof of Theorem \ref{['T1']}
  5. Preliminaries for Theorem \ref{['T2']}
  6. Proof of Theorem \ref{['T2']}

Key Result

Theorem 1.1

Assume that $\delta:\overline{\Omega}\to(0,1)$ is a continuous function and $h$ satisfies (H1)-(H5). Then, for every $\epsilon>0$, there exists $\Lambda_\epsilon>0$ such that Furthermore, if $\lambda\in(0,\Lambda_\epsilon)$, then the above solution is a local minimizer of the energy functional defined on $X.$

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • ...and 17 more