The extremal solution for the fractional Laplacian
نویسندگان
چکیده
منابع مشابه
The Extremal Solution for the Fractional Laplacian
We study the extremal solution for the problem (−∆)u = λf(u) in Ω, u ≡ 0 in R \ Ω, where λ > 0 is a parameter and s ∈ (0, 1). We extend some well known results for the extremal solution when the operator is the Laplacian to this nonlocal case. For general convex nonlinearities we prove that the extremal solution is bounded in dimensions n < 4s. We also show that, for exponential and power-like ...
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15 صفحه اولThe Pohozaev Identity for the Fractional Laplacian
In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem (−∆)u = f(u) in Ω, u ≡ 0 in R\Ω. Here, s ∈ (0, 1), (−∆) is the fractional Laplacian in R, and Ω is a bounded C domain. To establish the identity we use, among other things, that if u is a bounded solution then u/δ|Ω is C up to the boundary ∂Ω, where δ(x) = dist(x, ∂Ω). In the fractional Pohozaev identity, the func...
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ژورنال
عنوان ژورنال: Calculus of Variations and Partial Differential Equations
سال: 2013
ISSN: 0944-2669,1432-0835
DOI: 10.1007/s00526-013-0653-1