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We consider a semilinear elliptic problem with a nonlinear term which is theproduct of a power and the Riesz potential of a power. This family of equationsincludes the Choquard or nonlinear Schroedinger--Newton equation. We show thatfor some values of the parameters the equation does not have nontrivialnonnegative supersolutions in exterior domains. The same techniques yieldoptimal decay rates when supersolutions exists.

Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains