The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:473

Article

Colorado, Eduardo^1,2  Ortega, Alejandro¹ 

[1] Univ Carlos III Madrid, Dept Matemat, Avda Univ 30, Leganes 28911, Madrid, Spain

[2] UCM, UC3M, UAM, Inst Ciencias Matemat,ICMAT,CSIC, C Nicolas Cabrera 15, Madrid 28049, Spain

关键词: Fractional Laplacian;    Mixed boundary conditions;    Critical points;    Critical problems;    Semilinear problems;   

DOI  :  10.1016/j.jmaa.2019.01.006

来源: Elsevier

【 摘 要 】

In this work we study the existence of solutions to the critical Brezis-Nirenberg problem when one deals with the spectral fractional Laplace operator and mixed Dirichlet-Neumann boundary conditions, i.e., {(-Delta)(s)u = lambda u+u(2:-1), u > 0 in Omega, u = 0 on Sigma D; partial derivative u/partial derivative v = 0 on Sigma(N), where Omega C R-N is a regular bounded domain, 1/2 < s < 1, 2(s)(*); is the critical fractional Sobolev exponent, 0 <= lambda epsilon R, v is the outwards normal to partial derivative Omega, Sigma(D), Sigma(N) are smooth (N - 1)-dimensional submanifolds of partial derivative Omega such that Sigma(D) U Sigma(N) = partial derivative Omega , Sigma(D) boolean AND Sigma(N) = 0, and ED fl EAr = F is a smooth (N- 2)-dimensional submanifold of 812. (C) 2019 Elsevier Inc. All rights reserved.

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