Regularity results of nonlinear perturbed stable-like operators. (Q2216243)

The authors study the regularity property of nonlinear integro-differential elliptic operators \[\inf_{\beta}\sup_{\alpha}\int_{{R}^d}(u(x+y)+u(x-y)-2u(x))\frac{k_{\alpha\beta}(y)}{|y|^d}\,{d}y\] where \(k_{\alpha\beta}(y)\) is symmetric and satisfies \[(2-\alpha)\lambda\frac{1}{|y|^{\alpha}}\le k_{\alpha\beta}(y)\le\Lambda((2-\alpha)/|y|^{\alpha}+\varphi(1/|y|)),\,\,\,0<\lambda\le\Lambda,\] for \(\varphi\colon(0,\infty)\to(0,\infty)\) satisfying a weak upper scaling property with exponent \(\beta<\alpha\). Such operators do not have a global scaling property. Hölder regularity, Harnack inequality and boundary Harnack property of solutions of these operators are established.