Given a weak PDE $$ \langle b(u),v \rangle +\langle a(\nabla u), \nabla v\rangle=\langle f, v \rangle,\qquad v\in H^1_0 $$ where $b\in C(\mathbb{R})$ and $a\in C(\mathbb{R}^n,\mathbb{R}^n)$ grow slower than $|x|+1$, how can I find solutions to finite dimensional (Galerkin-) approximations (that is, testing only for a finite ONB $\{ v_k\}_{k=1,\dots,n}$). I am told to use Leray-Schauder, but the only idea i come up with to make this a fix-point equation, is to write $u=\sum_{i=1}^n a_i v_i$ and then add $a_k$ on both sides of the eqn. tested with $v_k$. Then I would have to show that for $\sigma\in [0,1]$ we have $$ \sigma(\langle b(u),v_k \rangle +h\langle a(\nabla u), \nabla v_k\rangle-\langle f, \forall v_k \rangle+a_k)=a_k , k=1\dots n $$ only for $|a|\leq M$. But I don't see how I would do this.
Any hint appreciated.