Small order asymptotics for nonlinear fractional problems

Víctor Hernández-Santamaría; Alberto Saldaña

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arXiv · 2108.00448

Small order asymptotics for nonlinear fractional problems

Víctor Hernández-Santamaría·Alberto Saldaña

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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.

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Víctor Hernández-Santamaría, Alberto Saldaña. 2021-08-01. Small order asymptotics for nonlinear fractional problems. https://arxiv.org/abs/2108.00448

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