Multiplicity conjecture for solutions with finitely many modes

Let $\mathbb{N}_0$ denote the set of nonnegative integers, and consider problem

, whose solutions may be decomposed into modes. A solution has **at most $M$ modes** if its mode decomposition contains no more than $M$ modes. **Multiplicity conjecture.** For each $M\in\mathbb{N}_0$, there \exists at least one distinct solution $u_M$ of problem

having at most $M$ modes.

This conjecture proposes a multiplicity result for the nonlocal elliptic problem, despite the absence of an energy functional. The supplied text gives no resolution or partial result concerning the conjecture.