Abstract
We take up in this paper the existence of positive continuous solutions for some nonlinear boundary value problems with fractional differential equation based on the fractional Laplacian \({(-\Delta _{|D})^{\frac{\alpha }{2}}}\) associated to the subordinate killed Brownian motion process \({Z_{\alpha }^{D}}\) in a bounded C1,1 domain D. Our arguments are based on potential theory tools on \({Z_{\alpha }^{D}}\) and properties of an appropriate Kato class of functions Kα(D).
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