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

TL;DR

This paper investigates the existence, non-existence, and uniqueness of solutions for nonlocal elliptic equations with the logarithmic Laplacian, introducing new identities and analyzing solution behaviors across different nonlinear growth regimes.

Contribution

It introduces Pohozaev's identity and Diaz-Saa inequality for logarithmic Laplacian problems and studies solution asymptotics depending on nonlinear growth and weight regularity.

Findings

Established Pohozaev's identity and Diaz-Saa inequality for the problem.

Proved convergence of solutions to Brézis-Nirenberg type problems.

Analyzed solution behavior for sublinear, superlinear, and critical nonlinearities.

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 D\'iaz-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 Br\'ezis-Nirenberg type and logistic-type limiting problem respectively involving the logarithmic Laplacian.