Nonexistence results for a class of fractional elliptic boundary value\n problems

2012/01/19 by Mouhamed Moustapha Fall, Fall, Mouhamed Moustapha, Tobias Weth +1 · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1201.4007

openalex publication_date 2012/01/19 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper we study a class of fractional elliptic problems of the form \n
Ds u= f(x,u)
quad
textrmin
O u=0
quad
textrmin
R^N
setminus
O,\nwhere s\∈(0,1). We prove nonexistence of positive solutions when O is\nstar-shaped and f is supercritical. We also derive a nonexistence result for\nsubcritical f in some unbounded domains. The argument relies on the method of\nmoving spheres applied to a reformulated problem using the Caffarelli-Silvestre\nextension citeCSilv of a solution of the above problem. The standard\napproach in the case s=1 using Pohozaev type identities does not carry over\nto the case 0<s<1 due to the lack of boundary regularity of solutions.\n

Citations

• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Cited by

• The Pohozaev identity for the fractional Laplacian
• The Dirichlet problem for the fractional Laplacian: regularity up to the\n boundary

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