Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth
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| Format: | Preprint |
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2025
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| _version_ | 1866915586817130496 |
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| author | Arora, Rakesh Giacomoni, Jacques Hajaiej, Hichem Vaishnavi, Arshi |
| author_facet | Arora, Rakesh Giacomoni, Jacques Hajaiej, Hichem Vaishnavi, Arshi |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26286 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth Arora, Rakesh Giacomoni, Jacques Hajaiej, Hichem Vaishnavi, Arshi Analysis of PDEs 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. |
| title | Sharp embeddings and existence results for Logarithmic $p$-Laplacian equations with critical growth |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2510.26286 |