Normalized solutions for the NLS equation with mixed fractional Laplacians and combined nonlinearities

Abstract

We look for normalized solutions to the nonlinear Schr\"{o}dinger equation with mixed fractional Laplacians and combined nonlinearities $$ \left\{\begin{array}{ll} (-\Delta)^{s_{1}} u+(-\Delta)^{s_{2}} u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u \ \text{in}\;{\mathbb{R}^{N}}, \\[0.1cm] \int_{\mathbb{R}^{N}}|u|^2\mathrm dx=a^2, \end{array} \right. $$ where $N\geq 2,\;0<s_2<s_1<1, \mu>0$ and $\lambda\in\mathbb R$ appears as an unknown Lagrange multiplier. We mainly focus on some special cases, including fractional Sobolev subcritical or critical exponent. More precisely, for $2<q<2+\frac{4s_2}{N}<2+\frac{4s_1}{N}<p<2_{s_1}^{\ast}:=\frac{2N}{N-2s_1}$, we prove that the above problem has at least two solutions: a ground state with negative energy and a solution of mountain pass type with positive energy. For $2<q<2+\frac{4s_2}{N}$ and $p=2_{s_1}^{\ast}$, we also obtain the existence of ground states. Our results extend some previous ones of Chergui et al. (Calc. Var. Partial Differ. Equ., 2023) and Luo et al. (Adv. Nonlinear Stud., 2022).