Regularity of nonlinear equations for fractional Laplacian

Xia, Aliang; Yang, Jianfu

doi:10.1090/S0002-9939-2013-11734-X

Regularity of nonlinear equations for fractional Laplacian
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by Aliang Xia and Jianfu Yang PDF

Proc. Amer. Math. Soc. 141 (2013), 2665-2672 Request permission

Abstract:

In this paper, we prove that any $H^s(\Omega )$ solution $u$ of the problem \begin{equation}(-\Delta )^s u=f(u) \textrm {in} \Omega , \quad u = 0 \textrm {on} \partial \Omega , \end{equation} belongs to $L^\infty (\Omega )$ for the nonlinearity of $f(t)$ being subcritical and critical. This implies that the solution $u$ is classical if $f(t)$ is $C^{1,\gamma }$ for some $0<\gamma <1$.

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