Saddle solutions for the critical Choquard equation

Abstract

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.