A Fractional Logistic-Type Elliptic Problem with Global Interactions: Existence, Local Uniqueness, and the Fractional-to-Local Limit
We study a fractional logistic-type elliptic problem on a bounded domain with homogeneous exterior conditions and linear and nonlinear integral interactions. The interaction kernels may be nonsymmetric and sign-changing, so the problem need not admit a variational formulation. In the symmetric competitive regime, coercivity yields a weak solution by global minimization. For arbitrary fixed values of the interaction parameter, explicit smallness conditions allow us to apply Schauder's fixed-point theorem without symmetry or sign restrictions on the kernels. Additional sign assumptions give nonnegative solutions. When the nonlinear exponent satisfies $q\geq2$, a contraction condition yields uniqueness in an invariant ball and convergence of the Picard iteration. Finally, as the fractional order approaches one, we prove subsequential convergence to a solution of the local Dirichlet problem, together with convergence of the fractional energies to the Dirichlet energy. Uniqueness of the local solution in the limiting ball gives convergence of the entire family.