Can some one help me find the reference which contains the proof of the following theorem :?

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Gianluca Argentini

Dear Shatha, see Haim Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. If in W1,p(Omega) you can establish a unicity result for the laplacian, then from your hypothesis u = 0 in the only solution. Gianluca

Shatha S Alhily

Dear Gianluca

Thank you so much for guiding me through that book , really it is a good reference.

Yes; you are right ...I am still reading in this subject.

Fabrizio Morlando

Hi, I want to point out that in the case p=1 the conclusion holds for the ball and it doesn't hold in general for less regular domain Omega.

See "Function Spaces and Partial Differential Equations: Contemporary Analysis"

by A. Taheri