Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations

By A. Capellà, J. D'avila, Louis Dupaigne, Yannick Sire. 2010

309 citations recorded by the source corpus.

We investigate stable solutions of elliptic equations of the type where n ≥ 2, s ∈ (0, 1), λ ≥0 and f is any smooth positive superlinear function. The operator (− Δ) s stands for the fractional Laplacian, a pseudo-differential operator of order 2s. According to the value of λ, we study the existence and regularity of weak solutions u.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/65b5d107044676f884becee304369017da1876dc

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In Applied Mathematics

In Computational Theory and Mathematics

In Control and Systems Engineering

Citations in the corpus

Equations involving fractional Laplacian operator: Compactness and application (2015, background): “We know from [4], see also [8], that for any u ∈ H 0 (Ω), let v ∈ H 1 0,L(CΩ) be the extension of u defined in (1.”
Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains (2014, methodology): “ 1, but it turns out to be impossible to write a clean equation due to the nonlocality of L, without passing through the extension technique by Caﬀarelli-Silvestre [15] in the elliptic case, see also [14, 16, 38]; as for the extension technique in the parabolic case, see [4] and [36, 37]. 6.1 The re”
Recent progress in the theory of Nonlinear Diffusion with Fractional Laplacian Operators (2014, unknown): “See also [42].”
The extremal solution for the fractional Laplacian (2013, background): “The closest result to ours was obtained by Capella-Dávila-Dupaigne-Sire [10].”
ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR NONLINEAR ELLIPTIC PROBLEMS WITH THE FRACTIONAL LAPLACIAN (2013, methodology): “[14] J. Dávila, M. del Pino, and Y. Sire, Non degeneracy of the bubble in the critical case for non local equations, Proc.”
Fractional diffusion with Neumann boundary conditions: the logistic equation (2012, background): “These arguments can be extended also to different powers of −∆, see [11].”
A General Fractional Porous Medium Equation (2011, unknown): “Caﬀarelli-Silvestre extension (3.3), restricted here to the upper half-cylinder CΩ = Ω × (0,∞), with null condition on the lateral boundary, ∂Ω × (0,∞), a construction considered in [12], [38], [10], [16]. Thus, w= E(|u|m−1u) satisﬁes (4.2) Lσw= 0, (x,y) ∈ CΩ,t>0, w= 0, x∈ ∂Ω,y>0,t>0”
On some critical problems for the fractional Laplacian operator (2011, methodology): “Therefore, again by [14], this time using Lemmas 2.”
Nonexistence results for a class of fractional elliptic boundary value problems (2012, methodology): “We point out that in [5,8, 12,15,32] a variant of the Caffarelli–Silvestre extension for solutions of (1.”
The Nehari manifold for fractional systems involving critical nonlinearities (2015, background): “We refer the reader to [3–5,10] and to the references therein.”
Radial symmetry of positive solutions to equations involving the fractional laplacian (2013, unknown): “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].”