Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions

By X. Cabré, Yannick Sire. 2011

294 citations recorded by the source corpus.

This paper, which is the follow-up to part I, concerns the equation (-Delta)(s)v + G'(v) = 0 in R-n, with s is an element of (0, 1), where (-Delta)(s) stands for the fractional Laplacian-the infinitesimal generator of a Levy process.; When n = 1, we prove that there exists a layer solution of the equation (i.e., an increasing solution with limits +/- 1 at +/-infinity) if and only if the potential G has only two absolute minima in [-1, 1], located at +/- 1 and satisfying G'(-1) = G'(1) = 0. Under the additional hypotheses G ''(-1) > 0 and G ''(1) > 0, we also establish its uniqueness and asymptotic behavior at infinity. Furthermore, we provide with a concrete, almost explicit, example of layer solution.; For n >= 1, we prove some results related to the one-dimensional symmetry of certain solutions-in the spirit of a well-known conjecture of De Giorgi for the standard Laplacian.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/9a438e0447d19c9dd3204cabcb3f675b794bf1c3

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Citations in the corpus

Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates (2010, methodology): “|→∞ v(|x|) = 0. (2.5) We can now state our main results. The next theorem provides a necessary condition —(2.7) and (2.8)— on the nonlinearity fto admit a layer solution in R. In our subsequent paper [4], this necessary condition will be proven to be also suﬃcient for the existence of a layer soluti”
Higher nonlocal problems with bounded potential (2014, methodology): “We cite the monograph [8] for related topics on variational methods adopted in this paper and [4, 5, 6] for recent nice results in the fractional setting.”
Nonlocal Diffusion and Applications (2015, methodology): “9) For the existence of such solution and its main properties see [75] and [15].”
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.”
Fractional Schrödinger–Poisson Systems with a General Subcritical or Critical Nonlinearity (2015, background): “2 [11] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians: I. Regularity, maximum principle and Hamiltonian estimates, Ann.”
ENERGY ESTIMATES AND 1-D SYMMETRY FOR NONLINEAR EQUATIONS INVOLVING THE HALF-LAPLACIAN (2010, result): “In general, it can be defined up to an additive constant c in the potential G(v)− c, but in this case, by the assumption (1.18) on G, we take c = 0.”
Radial symmetry of positive solutions to equations involving the fractional laplacian (2013, methodology): “On the other hand, using the local formulation of Caffarelli and Silvestre, Cabré and Sire [3] applied the sliding method to obtain symmetry results for nonlinear equations with”
On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, background): “onicity and rigidity of solutions to semilinear elliptic equations of fractional order in the whole space RN or in smooth bounded sets has attracted considerable attention in the last years, see e.g. [3,8,9,10,11,15,19,20,28,29,43]. In striking contrast, if is unbounded, but dierent from the whole s”
Traveling wave solutions of Allen–Cahn equation with a fractional Laplacian (2015, methodology): “If the regular Laplacian is replaced by fractional Laplacians (− ) , it is shown in [23] for s = 1/2 and [21,22] for s ∈ (0,1) that a standing wave solution exists in high dimensions by the s-harmonic extension (see also [80,81] and [78]).”
Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: A unified approach via fractional De Giorgi classes (2016, background): “See [13, 60, 10, 11, 12] for progresses in this direction and [32, 3, 56, 25, 30, 31] for the state-of-the-art results on the conjecture in the classical case, formally obtained by taking s = 1.”
Local and global minimizers for a variational energy involving a fractional norm (2011, unknown): “, it seems natural to study the solutions uof (3) that satisfy the following two conditions: ∂xnu(x) >0 for any x∈ Rn (4) and, possibly, lim xn→±∞ u(x′,xn) = ±1, for any x′ ∈ Rn−1. (5) We refer to [9,25,26,16,8,10,11] for several results in this direction. Here, by means of a technical variationo”