2025/02/11 by Huyuan Chen, Rui Chen, Chen, Huyuan +3
Mathematics · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.2502.07240
In this paper, we study critical semilinear nonlocal elliptic equations involving the logarithmic Schrödinger 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 \((I - Δ)ln\) with subcritical logarithmic nonlinearities. We then investigate a Brezis--Nirenberg-type problem involving the fractional pseudo-relativistic Schrödinger operator \((I - Δ)s\), 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 \(s → 0+\). 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.