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<p>In this paper, we derive a new <i>p</i>-Logarithmic Sobolev inequality and optimal continuous and compact embeddings into Orlicz-type spaces of the function space associated with the logarithmic <i>p</i>-Laplacian. As an application of these results, we study a class of Dirichlet boundary value problems involving the logarithmic <i>p</i>-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 <i>p</i>-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 <i>p</i>-Laplacian as the fractional parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s \rightarrow 0^+\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo stretchy="false">→</mo> <msup> <mn>0</mn> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

  • 期刊论文

  • 发表日期：

    2026年5月18日

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  • Rakesh Arora,
  • Jacques Giacomoni,
  • Hichem Hajaiej,
  • Arshi Vaishnavi

摘要

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 \(s \rightarrow 0^+\) s → 0 + .