Saddle solutions for the critical Choquard equation
Saddle solutions for the critical Choquard equation
复制标题
临界 Choquard 方程的鞍解
DOI：
10.1007/s00526-021-01919-5
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发表时间：
2021
影响因子：
2.1
通讯作者：
Xu Zhang
中科院分区：
文献类型：
--
作者：
Jiankang Xia;Xu Zhang
我们研究 Choquard 方程的鞍型节点解 $$\begin{aligned} -\Delta u = (I_\alpha *|u|^{\frac{N+\alpha }{N-2}} )|u|^{\frac{N+\alpha }{N-2}-2}u \quad \text { 在 }\;{\mathbb {R}}^N, \end{aligned}$$ 其中，是有序的 Riesz 势，指的是 Hardy–Littlewood–Sobolev 不等式的上临界指数。通过引入 Coxeter 的对称群，我们给出了一个统一的框架来构造具有指定对称节点结构的鞍解。为了克服由于缺乏紧性而产生的困难，我们对对称Palais-Smale序列的剖面分解进行了精细分析。这些结果进一步体现了 Choquard 方程的非局部性质，相比之下，对应的 Yamabe 方程不允许这种类型的节点解。
We study the saddle type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u = (I_\alpha *|u|^{\frac{N+\alpha }{N-2}} )|u|^{\frac{N+\alpha }{N-2}-2}u \quad \text { in }\;{\mathbb {R}}^N, \end{aligned}$$where,is the Riesz potential of orderandrefers to the upper critical exponent with respect to the Hardy–Littlewood–Sobolev inequality. By introducing the symmetric groups of Coxeter, we give a unified framework to construct saddle solutions with prescribed symmetric nodal structures. To overcome the difficulties arising from the lack of compactness, we give a fine analyzes on the profile decompositions of the symmetric Palais–Smale sequence. These results further feature the nonlocal nature of the Choquard equation, in contrast, the counterpart Yamabe equationcan not permit such type of nodal solutions.