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Let $(H,\langle \cdot , \cdot \rangle)$ a Hilbert space, $a:H\times H \rightarrow \mathbb R$ a bounded coercive bilineal form and $F:H\rightarrow \mathbb R$ linear and bounded. It is well-known that the Lax-Milgram theorem assures that there exists an unique $u \in H$ such that $$ a(u,v) = F(v) \quad \forall v \in H. $$ I was wondering, can we drop any hypothesis of the Lax-Milgram theorem just to guarantee that there exists a solution but it is not necessarily unique?

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