Saddle solutions for the Choquard equation

Abstract

We study the Choquard equation $$\begin{aligned} -\Delta u + u= (I_\alpha *|u|^p )|u|^{p-2}u \quad \text { in }\;{\mathbb {R}}^N \end{aligned}$$ where $$N\ge 2$$ , $$I_\alpha $$ is the Riesz potential of order $$\alpha \in (0, N)$$ and $$2\le p<\frac{N+\alpha }{N-2}$$ . For any given integer $$\ell \ge 2$$ , we construct a saddle type nodal solution with its $$2\ell $$ nodal domains meeting at the origin. Moreover, this equation admits nodal solutions of higher dimensional nodal structure with exactly $$2^k$$ nodal domains for each $$1\le k\le N$$ . This is a new phenomenon for the Choquard equation which is nonlocal in nature when compared with its local counterpart the nonlinear Schrodinger equation.