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 C 1,1 domain D. Our arguments are based on potential theory tools on \({Z_{\alpha }^{D}}\) and properties of an appropriate Kato class of functions K α (D).
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Dhifli, A., Mâagli, H. & Zribi, M. On the subordinate killed B.M in bounded domains and existence results for nonlinear fractional Dirichlet problems. Math. Ann. 352, 259–291 (2012). https://doi.org/10.1007/s00208-011-0642-7
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DOI: https://doi.org/10.1007/s00208-011-0642-7