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arXiv · 2411.15985

Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results

Rakesh Arora·Jacques Giacomoni·Arshi Vaishnavi
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Abstract

In this work, we study the existence, non-existence, and uniqueness results for nonlocal elliptic equations involving logarithmic Laplacian, and subcritical, critical, and supercritical logarithmic nonlinearities. The Poho\u zaev's identity and Díaz-Saa type inequality are proved, which are of independent interest and can be applied to a larger class of problems. Depending upon the growth of nonlinearities and regularity of the weight function, we study the small-order asymptotic of nonlocal weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that the least energy solutions of a weighted nonlocal problem with superlinear or sublinear growth converge to a nontrivial nonnegative least-energy solution of Brézis-Nirenberg type and logistic-type limiting problem respectively involving the logarithmic Laplacian.

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Rakesh Arora, Jacques Giacomoni, Arshi Vaishnavi. 2025-04-26. Nonlocal elliptic equations involving logarithmic Laplacian: Existence, non-existence and uniqueness results. https://arxiv.org/abs/2411.15985

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