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

and we look for solutions which satisfy prescribed mass

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.

我们研究带组合非线性的分数阶非线性Schrödinger方程的规范化解

我们正在寻找满足规定质量的解决方案

其中\（N \ ge 2，s \ in（0,1），\ mu \ in \ mathbb {R} \）和\（2 <q <p <2_s ^ * = 2N /（N-2s）\）。在\（q <p，a> 0 \）和\（\ mu \ in \ mathbb {R} \）中的不同假设下，我们证明了归一化解的存在性和不存在性。更具体地说，在纯粹的\（L ^ 2 \）-次临界情况下，我们借助最小能量值的单调性克服了紧凑性的不足，并获得了\（\ mu> 0 \）的基态解的存在。对于散焦情况\（\ mu <0 \），我们通过构造辅助函数来证明不存在的结果。我们强调，即使对于拉普拉斯算子，不存在的结果也是新的。在纯粹\（L ^ 2 \）-超临界情况，我们引入了一种纤维能量函数来获得Palais-Smale序列的有界性，并得到一个山-式解。在组合类型的情况下，我们构造不同的链接结构以获得鞍形解决方案。最后，我们指出，我们证明了均质非线性（\（\ mu = 0 \））的唯一性结果，该结果基于基态解的莫尔斯系数。