Normalized solutions to the fractional Schrödinger equations with combined nonlinearities

Abstract

We study the normalized solutions of the fractional nonlinear Schrödinger equations with combined nonlinearities

$$\begin{aligned} (-\Delta )^s u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2}u \quad \text{ in }~{\mathbb {R}}^N, \end{aligned}$$

and we look for solutions which satisfy prescribed mass

$$\begin{aligned} \int _{\mathbb {R}^N}|u|^2=a^2, \end{aligned}$$

where $N\ge 2,s\in (0,1),\mu \in \mathbb {R}$ and $2<q<p<2_s^*=2N/(N-2s)$. Under different assumptions on $q<p,a>0$ and $\mu \in \mathbb {R}$, we prove some existence and nonexistence results about the normalized solutions. More specifically, in the purely $L^2$-subcritical case, we overcome the lack of compactness by virtue of the monotonicity of the least energy value and obtain the existence of ground state solution for $\mu >0$. While for the defocusing situation $\mu <0$, we prove the nonexistence result by constructing an auxiliary function. We emphasis that the nonexistence result is new even for Laplacian operator. In the purely $L^2$-supercritical case, we introduce a fiber energy functional to obtain the boundedness of the Palais–Smale sequence and get a mountain-pass type solution. In the combined-type cases, we construct different linking structures to obtain the saddle type solutions. Finally, we remark that we prove a uniqueness result for the homogeneous nonlinearity ($\mu =0$), which is based on the Morse index of ground state solutions.