Existence and Convergence of Least-Energy Solutions Involving the Logarithmic Schr\"odinger Operator

TL;DR

This paper investigates the existence, uniqueness, and convergence of least-energy solutions to nonlocal logarithmic Schrödinger equations, employing variational methods and asymptotic analysis.

Contribution

It establishes the existence and uniqueness of solutions involving the logarithmic Schrödinger operator and proves the convergence of least-energy solutions as the fractional parameter tends to zero.

Findings

Existence and regularity of solutions for the logarithmic Schrödinger operator.

Existence of least-energy solutions for fractional pseudo-relativistic equations.

Convergence of solutions to a limiting problem as the fractional parameter approaches zero.

Abstract

In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schr\"odinger operator and its fractional pseudo-relativistic counterpart, both arising in quantum models with nonlocal and relativistic effects. We first establish the existence, uniqueness, and regularity of weak solutions to equations involving the logarithmic operator \((I - \Delta)^{\ln}\) with subcritical logarithmic nonlinearities. We then investigate a Brezis--Nirenberg-type problem involving the fractional pseudo-relativistic Schr\"odinger operator \((I - \Delta)^s\), and prove the existence of least-energy solutions under both subcritical and critical nonlinearities. In particular, we show that these least-energy solutions converge, up to a subsequence, to a nontrivial least-energy solution of the limiting problem as \(s \to 0^+\). Our approach relies on variational methods, including the geometry of the Nehari manifold, uniform positive lower bounds, mountain-pass structure, the Palais--Smale condition, and delicate asymptotic analysis.