Publication: The Brezis-Nirenberg problem for the fractional Laplacian with mixed Dirichlet-Neumann boundary conditions
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2019-05-15
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Elsevier
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Abstract
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.
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Fractional Laplacian, Mixed boundary conditions, Critical points, Critical problems, Semilinear problems
Bibliographic citation
Colorado, E., Ortega, A. (2019). The Brezis–Nirenberg problem for the fractional Laplacian with mixed Dirichlet–Neumann boundary conditions. Journal of Mathematical Analysis and Applications, 473(2), 1002–1025