Regularity of nonlinear equations for fractional Laplacian (Q2839350)

Let \(\Omega \subset \mathbb R^N, \) \(N \geq 2\) a smooth bounded domain. The aim of the authors is to prove that any \(u\in H^s(\Omega), \) \(0<s<1, \) solution of the following problem involving fractional Laplacian operators NEWLINE\[NEWLINE(-\Delta)^s u = f(u) \text{ in }\Omega, NEWLINE\]NEWLINE NEWLINE\[NEWLINEu= 0 \text{ on } \partial \Omega,NEWLINE\]NEWLINE belongs to \(L^\infty(\Omega), \) when the nonlinearity of \(f(t)\) is subcritical or critical.NEWLINENEWLINEThis implies that the solution \(u\) is classical if \(f(t)\) such that NEWLINE\[NEWLINE |f(t)|\leq C (1+|t|^p), 1<p<\frac{N-2s}{N+2s}, NEWLINE\]NEWLINE is \(C^{1,\gamma}\) for some \(0 < \gamma < 1. \)NEWLINENEWLINERecently, the above problem has been studied by many authors, e.g., X. Cabré, A. Capella, Y. Sire, J. Tan, L. Farelli, L. Silvestre, J. D'Avila, L. Dupaigne. The existence of solutions and also various properties of the solutions has been considered. In particular, the regularity of the weak solutions was proved, as in the present paper. Let us point out that the restriction of the exponent \(p\) is natural in the sense of variational methods.