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arXiv · 2110.09941

Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls

Guillaume Penent·Nicolas Privault
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Abstract

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension $d\geq 1$. Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all $s\in (0,1)$, and our existence results hold for sufficiently small exterior conditions and nonlinearity coefficients. In comparison with existing approaches, we consider a wide class of polynomial nonlinearities without imposing upper bounds on their maximal degree or number of terms. Numerical illustrations are provided in large dimensions.

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Guillaume Penent, Nicolas Privault. 2021-10-19. Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls. https://arxiv.org/abs/2110.09941

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