You might be interested in reading Chang's "Infinite Dimensional Morse Theory" or Jost's "Riemannian Geometry and Geometric Analysis". Moreover, I guess it would be useful to give you an example: there is a result due to Bahri and Coron that states as follows: if $\Omega \subset \mathbb{R}^N$, $N\ge 3$ is a smooth bounded domain, you can find a solution of
$\begin{cases}-\Delta u = u^{\frac{N+2}{N-2}}&\text{in }\Omega\\ u > 0 & \text{in }\Omega\\ u = 0&\text{on }\partial \Omega,\end{cases}$
provided that the domain $\Omega$ is "topologically nontrivial", that is if the homology group $H_q(\Omega, \mathbb{Z}_2) \ne 0$ for some $q > 0$. Of course, there are a lot of more recent works in the study of pdes which involves techniques from algebraic topology.