Nonexistence results for a class of fractional elliptic boundary value problems

By M. Fall, Tobias Weth. 2012

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Abstract In this paper we study a class of fractional elliptic problems of the form { ( − Δ ) s u = f ( x , u ) in Ω , u = 0 in R N ∖ Ω , where s ∈ ( 0 , 1 ) . We prove nonexistence of positive solutions when Ω is star-shaped and f is supercritical. We also derive a nonexistence result for subcritical f in some unbounded domains. The argument relies on the method of moving spheres applied to a reformulated problem using the Caffarelli–Silvestre extension (Caffarelli and Silvestre (2007) [11] ) of a solution of the above problem.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c022a9bc410b2b213324fd10d5f3b08059cbd1da

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Unique Continuation Property and Local Asymptotics of Solutions to Fractional Elliptic Equations (2013, unknown): “Some computations (see also [10]) show that ∫ B 1 t1−2s|∇w|2dz + (N − 2s) ∫ S 1 t1−2sw2dS = ∫”
A Global Compactness type result for Palais-Smale sequences in fractional Sobolev spaces (2014, background): “The authors would like to thank Marco Squassina for having pointed out to them the paper [5].”
Monotonicity and nonexistence results for some fractional elliptic problems in the half-space (2013, methodology): “The argument of [21], relying on the method of moving spheres, does not apply under the assumptions of Theorem 1.”
Liouville Type Theorems for PDE and IE Systems Involving Fractional Laplacian on a Half Space (2017, background): “Comparing the results in [12, 13, 26] with our main results on system (2) (Corollary 4), one should observe two essential differences.”
A critical fractional equation with concave-convex power nonlinearities (2013, background): “Previous works related to the operator deﬁned in (1.2), or by a more general kernel, can be found in [16,23, 30,34,35,37,38,41–43 Problems similar to (P λ ) have been also studied in the local setting with different elliptic operators. far as we know, the ﬁrst example in this direction was given in…”
The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary (2012, background): “Its proof requires the precise regularity theory up to the boundary developed in the present paper (see Corollary 1.6 below).”
On Fractional Laplacians (2013, background): “In this case both the operators (−∆Ω) s N and (−∆Ω) s D were considered in many articles on semilinear equations, see for instance [2, 9, 11, 12, 13, 14, 17], and compared in [8], [15].”
The Pohozaev Identity for the Fractional Laplacian (2012, unknown): “However, in [12] they do not need to assume any regularity on f with respect to x .”
The extremal solution for the fractional Laplacian (2013, methodology): “The moving planes method has been applied to problems involving the fractional Laplacian by different authors; see for example [11, 1, 17].”
Liouville type theorems for nonlinear elliptic equations and systems involving fractional Laplacian in the half space (2015, unknown): “6 of [15] in the case N+2α N−2α < p < (N−1)+2α (N−1)−2α , see also Theorem 1.”