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Abstract

In this work, we study the non-local analogue of Brezis-Nirenberg and logistic type elliptic equations involving the logarithmic Laplacian and critical logarithmic non-linearity with superlinear-subcritical perturbation. In the first part of this work, we derive new sharp, continuous and compact embeddings of nonlocal Sobolev spaces (of order zero) into Orlicz type spaces. As an application of these embeddings and variational analysis as carried out in \cite{Angeles-Saldana-2023, Santamaria-Saldana-2022}, we prove the existence of a least energy weak solution of the Brezis-Nirenberg and logistic type problem involving the logarithmic Laplacian. For the uniqueness of solution, we prove a new D\'iaz-Saa type inequality, which is of independent interest and can be applied to a larger class of problems. In the second part of the work, depending upon the growth of non-linearity and regularity of the weight function, we study the small-order asymptotic of non-local weighted elliptic equations involving the fractional Laplacian of order $2s.$ We show that least energy solutions of a weighted non-local fractional problem with superlinear or sublinear type non-linearity converge to a non-trivial, non-negative least energy solution of a Brezis-Nirenberg type or logistic-type problem, respectively, involving the logarithmic Laplacian.