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

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

Automated summary

In this work, we prove the uniqueness and nondegeneracy of solutions to the nonlocal linear equation in the presence of a positive or odd solution, by utilizing a nonlocal Liouville-type method. This is the first work to establish such results in the nonlocal framework without using the Caffarelli-Silvestre extension technique.