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

TL;DR

This paper establishes the existence of multiple normalized solutions for a nonlinear Schrödinger equation involving mixed fractional Laplacians and combined nonlinearities, covering subcritical and critical cases, extending previous results.

Contribution

It introduces new existence results for normalized solutions in fractional Schrödinger equations with mixed operators and nonlinearities, including ground states and mountain pass solutions.

Findings

Existence of at least two solutions for certain nonlinear exponents.

Presence of a ground state with negative energy.

Existence of a mountain pass type solution with positive energy.

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).