Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates

By X. Cabré, Yannick Sire. 2010

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This is the first of two articles dealing with the equation $(-\Delta)^{s} v= f(v)$ in $\mathbb{R}^{n}$, with $s\in (0,1)$, where $(-\Delta)^{s}$ stands for the fractional Laplacian ---the infinitesimal generator of a L\'evy process. This equation can be realized as a local linear degenerate elliptic equation in $\mathbb{R}^{n+1}_+$ together with a nonlinear Neumann boundary condition on $\partial \mathbb{R}^{n+1}_+=\mathbb{R}^{n}$. In this first article, we establish necessary conditions on the nonlinearity $f$ to admit certain type of solutions, with special interest in bounded increasing solutions in all of $\mathbb{R}$. These necessary conditions (which will be proven in a follow-up paper to be also sufficient for the existence of a bounded increasing solution) are derived from an equality and an estimate involving a Hamiltonian ---in the spirit of a result of Modica for the Laplacian. In addition, we study regularity issues, as well as maximum and Harnack principles associated to the equation.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9f95017dec88363afa58ddae2400d10a4ee68a15

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Citations in the corpus

On a fractional Nirenberg problem, part I: blow up analysis and compactness of solutions (2011, methodology): “9) without the lower order term b, has been obtained earlier in [19] using a different method.”
Radial symmetry of positive solutions to equations involving the fractional laplacian (2013, positive): “Various regularity issues for fractional elliptic equations has been studied by Cabré and Sire [2], Caffarelli and Silvestre [6], Capella, Dávila, Dupaigne and Sire [7], Ros-Oton and Serra [22] and Silvestre [25].”
Local and global minimizers for a variational energy involving a fractional norm (2011, positive): “(5) We refer to [9,25,26,16,8,10,11] for several results in this direction.”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, methodology): “Moreover, in this analysis, the regularity of the principal eigenfunction and a Harnack type inequality [16, 17, 61] for some non negative solution of (1.”
Uniqueness of Radial Solutions for the Fractional Laplacian (2013, methodology): “Inspired by the work of Cabré and Sire [9] (see also [10] for earlier work in the case s D 1 2 ) on layer and radial solutions of nonlinear equations of the form .”
A Harnack inequality for fractional Laplace equations with lower order terms (2010, methodology): “eeden [10]. If σ = 1 2, the result is due to Han and Li [7]. After we complete our manuscript, we observe that Theorem 1.2, the Harnack inequality for b ≡ 0, has been shown recently by Cabre and Sire [2] through making even extension and using the result of Fabes, Kenig and Serapioni [5]. But our pr”
On the Evolutionary Fractional p-Laplacian (2015, background): “Literature The fractional Laplacian as a nonlocal generalization of the Laplace operator has been studied in classical monographs such as [23, 31] as well as in very recent articles such as [8, 9, 20] and many others.”
Ground state of scalar field equations involving a fractional Laplacian with general nonlinearity (2013, background): “By the maximum principle for the fractional Laplacian (see [12]), a solution of (1.”
Existence results for fractional p-Laplacian problems via Morse theory (2014, background): “ions. 1 2 A. IANNIZZOTTO, S. LIU, K. PERERA, AND M. SQUASSINA examined by Caffarelli & Silvestre [8], Ros Oton & Serra [37–39]. The corresponding equation in RN is studied by Cabré & Sire [4,5]. Although the fractional Laplacian operator (−∆)s, and more generally pseudodiﬀerential operat”