Existence and dynamics of normalized solutions to nonlinear Schr\"{o}dinger equations with mixed fractional Laplacians

Abstract

In this paper, we are concerned with the existence and dynamics of solutions to the equation with mixed fractional Laplacians $$ (-\Delta)^{s_1} u +(-\Delta)^{s_2} u + \lambda u=|u|^{p-2} u $$ under the constraint $$ \int_{\R^N} |u|^2 \, dx=c>0, $$ where $N \geq 1$, $0<s_2<s_1<1$, $2+ \frac {4s_1}{N} \leq p< \infty $ if $N \leq 2s_1$, $2+ \frac {4s_1}{N} \leq p<\frac{2N}{N-2s_1}$ if $N >2s_1$, $\lambda \in \R$ appearing as Lagrange multiplier is unknown. The fractional Laplacian $(-\Delta)^s$ is characterized as $\mathcal{F}((-\Delta)^{s}u)(\xi)=|\xi|^{2s} \mathcal{F}(u)(\xi)$ for $\xi \in \R^N$, where $\mathcal{F}$ denotes the Fourier transform. First we establish the existence of ground state solutions and the multiplicity of bound state solutions. Then we study dynamics of solutions to the Cauchy problem for the associated time-dependent equation. Moreover, we establish orbital instability of ground state solutions.