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

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.