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Normalized solutions to the fractional Schrödinger equations with combined nonlinearities
Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-08-13 , DOI: 10.1007/s00526-020-01814-5
Haijun Luo , Zhitao Zhang

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.



中文翻译:

分数阶Schrödinger方程组合非线性的归一化解

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

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

我们正在寻找满足规定质量的解决方案

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

其中\(N \ ge 2,s \ in(0,1),\ mu \ in \ mathbb {R} \)\(2 <q <p <2_s ^ * = 2N /(N-2s)\)。在\(q <p,a> 0 \)\(\ mu \ in \ mathbb {R} \)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。

更新日期:2020-08-14
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