Uniqueness for linear integro-differential equations in the real line and applications

Abstract In this work we prove the uniqueness of solutions to nonlocal linear equation $$L \varphi - c(x)\varphi = 0$$ L φ - c ( x ) = 0 in $$\mathbb {R}$$ R , where L is an elliptic integro-differential operator, presence a positive solution or odd vanishing only at zero. As application, deduce nondegeneracy layer (bounded and monotone solutions) semilinear problem u f(u)$$ u f when nonlinearity Allen–Cahn type. To our knowledge, first such results are proven framework Caffarelli–Silvestre extension technique not available. Our proofs based on Liouville-type method developed by Hamel, Ros-Oton, Sire, Valdinoci for nonlinear problems dimension two.