Ground states for pseudo-relativistic Schrödinger equations with logarithm-law: qualitative properties and nonrelativistic limit

Abstract

In this paper, we study the pseudo-relativistic Schrödinger equation with logarithmic nonlinearity \begin{equation*} \left(\sqrt{-c^{2}Δ+m^{2}c^{4}}-mc^{2}\right)u+μu=u\log|u| \quad\text{in }\mathbb{R}^{N}. \end{equation*} The existence and qualitative properties of action ground states are established, by means of a power-law approximation. We then prove that these states converge to the Gausson in $H^1({\mathbb{R}}^N)$ as $c\to \infty$. For sufficiently large $c$, we further obtain uniqueness of action ground states up to translation and sign change, as well as their nondegeneracy. Finally, we prove an exact correspondence between action and energy ground states, thereby extending the preceding results to energy ground states at arbitrary prescribed mass.