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I've already checked some functional analysis books and most of them don't cover Sobolev spaces

I know some measure theory and Hilbert spaces and would like to learn about sobolev spaces and mainly applications of the Lax–Milgram theorem

like if there's a book that treat the Dirichlet problem and/or the Neumann's problem it would be great.

thanks !

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Brezis' book discuses Sobolev spaces in detail and then treats the applications you're looking for.

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I would advice 3 different books:

1) "A first course in Sobolev Spaces", G. Leoni: a very long, precise and detailed book mainly focused on $W^{1,p}$ spaces, good for a first but theoretical approach (I think applications are not discussed here).

2) "Partial differential equations in action: from modelling to theory", S. Salsa: another really good and comprehensive book, more focused on application (PDEs); briefly introduces Sobolev Spaces (I think only the Hilbert case $p=2$) and related functional analysis tools to study PDEs problems.

3) "Partial differential equations" L. Evans: a super classic book, again Sobolev Spaces are introduced and immediately applied to PDEs and calculus of variations. Maybe a little more complicated than Salsa's.

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