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

@inproceedings{Penent2021ExistenceOS, title={Existence of solutions for nonlinear elliptic PDEs with fractional Laplacians on open balls}, author={Guillaume Penent and Nicolas Privault}, year={2021} }

We prove the existence of viscosity solutions for fractional semilinear elliptic PDEs on open balls with bounded exterior condition in dimension d ≥ 1. Our approach relies on a tree-based probabilistic representation based on a (2s)-stable branching processes for all s ∈ (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…

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## References

SHOWING 1-10 OF 38 REFERENCES

Existence and probabilistic representation of the solutions of semilinear parabolic PDEs with fractional Laplacians

- Mathematics
- 2021

We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on Rd with polynomial nonlinearities in (u,∇u), using a tree-based probabilistic representation. This…

Nonexistence results for a class of fractional elliptic boundary value problems

- Mathematics
- 2012

Abstract In this paper we study a class of fractional elliptic problems of the form { ( − Δ ) s u = f ( x , u ) in Ω , u = 0 in R N ∖ Ω , where s ∈ ( 0 , 1 ) . We prove nonexistence of positive…

Mountain Pass solutions for non-local elliptic operators

- Mathematics
- 2012

Abstract The purpose of this paper is to study the existence of solutions for equations driven by a non-local integrodifferential operator with homogeneous Dirichlet boundary conditions. These…

Existence of a positive solution for a class of fractional elliptic problems in exterior domains involving critical growth

- Mathematics
- 2022

Abstract In this paper we show existence of positive solutions for a class of problems involving the fractional Laplacian in exterior domain and the nonlinearity with critical growth. We prove the…

WEAK AND VISCOSITY SOLUTIONS OF THE FRACTIONAL LAPLACE EQUATION

- Mathematics
- 2014

Aim of this paper is to show that weak solutions of the following fractional Laplacian equation ( ( ) s u = f in u = g in R n n are also continuous solutions (up to the boundary) of this problem in…

Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

- Medicine, Computer ScienceEntropy
- 2021

Conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type are given and the optimality of some assumptions is argued.

Nonlocal elliptic equations in bounded domains: a survey

- Physics, Mathematics
- 2015

In this paper we survey some results on the Dirichlet problem ( Lu = f in u = g in R n n for nonlocal operators of the form Lu(x) = PV Z Rn u(x) u(x + y) K(y)dy: We start from the very basics,…

Finite difference methods for fractional Laplacians

- Mathematics
- 2016

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open…

The Dirichlet problem for nonlocal operators

- Mathematics
- 2013

In this note we set up the elliptic and the parabolic Dirichlet problem for linear nonlocal operators. As opposed to the classical case of second order differential operators, here the “boundary…

The Brownian snake and solutions of Δu=u2 in a domain

- Mathematics
- 1995

SummaryWe investigate the connections between the path-valued process called the Brownian snake and nonnegative solutions of the partial differential equation Δu=u2 in a domain of ℝd. In particular,…