Infinitely many solutions to a conformally invariant elliptic equation with Choquard-type nonlinearity

Abstract: The existence of an unbounded sequence of solutions to a conformally invariant elliptic equation having nonlocal critical-power nonlinearity is established. The primary obstacle to establishing existence of solutions is the failure of compactness in the Sobolev embedding. To overcome this obstacle, the problem under consideration is lifted to an equivalent problem on the standard sphere so that the symmetries of the sphere can be leveraged. Two classes of symmetries are considered and for each class of symmetries, an unbounded sequence of solutions to the lifted problem with the prescribed symmetries is produced. One class of symmetries always exists and the corresponding solutions are guaranteed to be sign-changing whenever a suitable relationship between the dimension and the nonlocality parameter holds. The other class of symmetries need not always exist but when it exists, the corresponding solutions are guaranteed to be sign-changing.