Small order asymptotics for nonlinear fractional problems

Abstract

We study the limiting behavior of solutions to boundary value nonlinear problems involving the fractional Laplacian of order 2s when the parameter s tends to zero. In particular, we show that least-energy solutions converge (up to a subsequence) to a nontrivial nonnegative least-energy solution of a limiting problem in terms of the logarithmic Laplacian, i.e. the pseudodifferential operator with Fourier symbol \(\ln (|\xi |^2)\). These results are motivated by some applications of nonlocal models where a small value for the parameter s yields the optimal choice. Our approach is based on variational methods, uniform energy-derived estimates, and the use of a new logarithmic-type Sobolev inequality.