Small order asymptotics for nonlinear fractional problems

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