Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2021-02-10 , DOI: 10.1007/s00526-021-01919-5 Jiankang Xia , Xu Zhang
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 \(N\ge 3\), \(I_\alpha \) is the Riesz potential of order \(\alpha \in (0, N)\) and \(\frac{N+\alpha }{N-2}\) refers 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 equation \(-\Delta u=|u|^{\frac{4}{N-2}}u\) can not permit such type of nodal solutions.
中文翻译：
临界Choquard方程的鞍形解
我们研究Choquard方程的鞍型节点解
$$ \ begin {aligned}-\ Delta u =（I_ \ alpha * | u | ^ {\ frac {N + \ alpha} {N-2}}）| u | ^ {\ frac {N + \ alpha} {N -2} -2} u \ quad \ text {in} \; {\ mathbb {R}} ^ N，\ end {aligned} $$其中\（N \ ge 3 \），\（I_ \ alpha \）是\（\ alpha \ in（0，N）\）和\（\ frac {N + \ alpha} {N-2 } \）是关于Hardy–Littlewood–Sobolev不等式的上限临界指数。通过介绍Coxeter的对称群，我们给出了一个统一的框架，以按规定的对称节点结构构造鞍形解。为了克服由于缺乏紧密性而引起的困难，我们对对称的Palais-Smale序列的轮廓分解进行了精细分析。这些结果还具有Choquard方程的非局部性质，相反，对应的Yamabe方程\（-\ Delta u = | u | ^ {\ frac {4} {N-2}} u \） 不允许这种类型的节点解决方案。
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