## Normalized solutions to the fractional Schrödinger equations with combined nonlinearities.(English)Zbl 1445.35307

Summary: We study the normalized solutions of the fractional nonlinear Schrödinger equations with combined nonlinearities $(-\Delta )^s u=\lambda u+\mu |u|^{q-2}u+|u|^{p-2} u \quad \text{ in }\mathbb{R}^N,$ and we look for solutions which satisfy prescribed mass $\int_{\mathbb{R}^N}|u|^2=a^2,$ where $$N\geq 2$$, $$s\in (0,1)$$, $$\mu \in \mathbb{R}$$ 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.

### MSC:

 35R11 Fractional partial differential equations 26A33 Fractional derivatives and integrals 35J61 Semilinear elliptic equations
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### References:

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