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Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth

Rakesh Arora, Jacques Giacomoni, Hichem Hajaiej, Arshi Vaishnavi

TL;DR

This work develops a nonlinear variational framework for the logarithmic $p$-Laplacian, introducing a new $p$-logarithmic Sobolev inequality and establishing optimal continuous and compact embeddings of the associated energy space into Orlicz-type spaces. These embeddings enable handling of critical logarithmic nonlinearities and yield existence and, in some regimes, uniqueness of least-energy solutions for Brezis–Nirenberg-type and logistic-type problems driven by the logarithmic operator. The authors further study the small-order limit of weighted nonlocal problems involving the fractional $p$-Laplacian, showing convergence of least-energy solutions to limiting problems governed by $L_{ riangle_p}$ as $s\to0^+$. Collectively, the results extend known linear theories to a broader nonlinear variational setting and provide nonlinear analogues of prior log-Sobolev and embedding results, with potential applications in nonlocal models and logarithmic-type nonlinear phenomena.

Abstract

In this paper, we derive a new $p$-Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the logarithmic $p$-Laplacian. As an application of these results, we study a class of Dirichlet boundary value problems involving the logarithmic $p$-Laplacian and critical growth nonlinearities perturbed with superlinear-subcritical growth terms. By employing the method of the Nehari manifold, we prove the existence of a nontrivial weak solution. Lastly, we conduct an asymptotic analysis of a weighted nonlocal, nonlinear problem governed by the fractional $p$-Laplacian with superlinear or sublinear type non-linearity, demonstrating the convergence of least energy solutions to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic $p$-Laplacian as the fractional parameter $s \to 0^+$. The findings in this work serve as a nonlinear analogue of the results reported in \cite{Angeles-Saldana, Arora-Giacomoni-Vaishnavi, Santamaria-Saldana}, thereby extending their scope to a broader variational framework.

Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth

TL;DR

This work develops a nonlinear variational framework for the logarithmic -Laplacian, introducing a new -logarithmic Sobolev inequality and establishing optimal continuous and compact embeddings of the associated energy space into Orlicz-type spaces. These embeddings enable handling of critical logarithmic nonlinearities and yield existence and, in some regimes, uniqueness of least-energy solutions for Brezis–Nirenberg-type and logistic-type problems driven by the logarithmic operator. The authors further study the small-order limit of weighted nonlocal problems involving the fractional -Laplacian, showing convergence of least-energy solutions to limiting problems governed by as . Collectively, the results extend known linear theories to a broader nonlinear variational setting and provide nonlinear analogues of prior log-Sobolev and embedding results, with potential applications in nonlocal models and logarithmic-type nonlinear phenomena.

Abstract

In this paper, we derive a new -Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the logarithmic -Laplacian. As an application of these results, we study a class of Dirichlet boundary value problems involving the logarithmic -Laplacian and critical growth nonlinearities perturbed with superlinear-subcritical growth terms. By employing the method of the Nehari manifold, we prove the existence of a nontrivial weak solution. Lastly, we conduct an asymptotic analysis of a weighted nonlocal, nonlinear problem governed by the fractional -Laplacian with superlinear or sublinear type non-linearity, demonstrating the convergence of least energy solutions to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic -Laplacian as the fractional parameter . The findings in this work serve as a nonlinear analogue of the results reported in \cite{Angeles-Saldana, Arora-Giacomoni-Vaishnavi, Santamaria-Saldana}, thereby extending their scope to a broader variational framework.

Paper Structure

This paper contains 15 sections, 38 theorems, 296 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Variational framework
  4. Main results
  5. Embeddings into Orlicz spaces
  6. p-Logarithmic Sobolev inequality
  7. Continuous and compact embeddings
  8. Sharpness of embeddings
  9. Logarithmic p-Laplacian equations with critical growth
  10. Properties of the energy functional
  11. The Brezis-Nirenberg type problem
  12. The Logistic type problem
  13. Small order asymptotics of the p-fractional Laplacian problem
  14. The superlinear case
  15. The sublinear case

Key Result

Theorem 2.5

(p-Logarithmic Sobolev inequality) For $u \in X_{L_{\Delta_p}}(\Omega),$ it holds that where $k_0$ is a constant depending upon $N$ and $p.$

Theorems & Definitions (70)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 60 more