Monotonicity and nonexistence results for some fractional elliptic problems in the half-space
By M. Fall, Tobias Weth. 2013
65 citations recorded by the source corpus.
We study a class of fractional elliptic problems of the form (−Δ)su = f(u) in the half-space ℝ+N := {x ∈ ℝN:x 1 > 0} with the complementary Dirichlet condition u ≡ 0 in ℝN∖ℝ +N. Under mild assumptions on the nonlinearity f, we show that bounded positive solutions are increasing in x1. For the special case f(u) = uq, we deduce nonexistence of positive bounded solutions in the case where q > 1 and q < N−1+2s N−1−2s if N ≥ 1 + 2s. We do not require integrability assumptions on the solutions we study.
Subfields: Applied Mathematics, Computational Theory and Mathematics
View Paper: https://www.semanticscholar.org/paper/f866651e0c7c6aa7a6c07e042b815dfd5fc5de4d
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In Applied Mathematics
In Computational Theory and Mathematics
Citations in the corpus
- Overdetermined problems with fractional Laplacian (2013, methodology): “The moving plane method have been also employed within the fractional framework in [3, 9, 17, 21, 24, 29].”
- Entire $s$-harmonic functions are affine (2014, unknown): “ − s)π−N/24s Γ(N 2 +s) Γ(2−s) . For u ∈ L 1 s, the expression (−∆)su defines a distribution and if (−∆)su = 0 in some open set then u is equivalent to a smooth function there, see for instance [3] and [5] (this can be also seen from the fact that (−∆)s is a pseudo-differential operator which is hypoel”
- Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space (2017, methodology): “Reichel, W., Weth, T.: A prior bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems.”
- On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results (2016, background): “In all the articles [2, 25, 35] the monotonicity is used to derive non-existence results for nondecreasing nonlinearities f , which is a complementary situation with respect to the one considered here.”