Existence and Limiting Profiles of Normalized Travelling Wave Solutions for the Pseudo-Relativistic Schr\"{o}dinger Equation with Logarithmic Nonlinearity

Abstract

We study the existence and asymptotic behaviour of normalized solutions to the following pseudo-relativistic Schr\"{o}dinger equation with logarithmic nonlinearity \[ (\sqrt{-\Delta+m^2 }-m )u+i(v\cdot \nabla )u+\lambda u = u\log|u|^2+|u|^{p-2}u, \qquad \text{in }\mathbb{R}^N, \] under the mass constraint \[ \|u\|_2^2=a, \] where $m,a>0$, $2<p\le\frac{2N}{N-1}$ with $N\ge 2$, $v\in \mathbb{R}^N$ is the travelling velocity with $|v|<1$, and $\lambda\in\mathbb{R}$ appears as Lagrange multiplier, as minima of the corresponding energy on the constraint. By applying variational method, we first provide a complete classification of the existence and nonexistence of such minima. In particular, for the mass-critical case $p=2+\frac{2}{N}$, we show that there exists a constant $a^\ast_v$ which is a threshold for the existence. Based on this, we analyse the blow-up behaviour of such minimizers as $a$ approaches $a^\ast_v$ from below. Finally, we investigate the limiting profiles of minimizers to problem when $\lim\limits_{n\to\infty}a_n=a_0\in(0,+\infty)$ with $\{a_n\}\subset(0,+\infty)$ in the mass-subcritical case $2<p<2+\frac{2}{N}$ and $\lim\limits_{n\to\infty}a_n= a_0\in(0,a^\ast_v)$ with $\{a_n\}\subset(0,a^\ast_v)$ in the mass-critical case $p=2+\frac{2}{N}$, respectively.