As stated in the comment section this is a vast topic. I will just name a few directions that you can explore.
-Degree theory: in finite dimension a map $f:M^n\to N^n$ between closed manifolds of the same dimension has a degree which can be defined as a signed sum of the points in the fiber $f^{-1}(y_0)$ of a regular value $y_0$. This enjoys several properties, for example it does not really depend on $y_0$ and it depends on $f$ only up to homotopy.
This can be generalized (under suitable assumptions)to PDEs once we realize the solution set of a PDE as the zero locus of a map $F:X\to Y$ where $X, Y$ are Banach spaces/manifolds.
Then if $F$ has positive degree it means that the fiber is non-empty, from this one can conclude the existence of solutions of a perturbation of $F$ or of $F$ itself if $0$ is a regular value.
A book to read about this is Deimling "Nonlinear functional analysis".
-Morse theory: this has been first applied by Marston Morse to prove results about the existence of geodesics on closed manifolds.
We have a functional $E:X\to \mathbb R$ (the energy, e.g. the length of a curve, $X$ a set of curves) and we are interested critical points of $E$, i.e. solutions to the equation $\nabla E (x) = 0 $. For example the geodesic equation can be formalized in this way (see Critical Curves of the Energy Functional are Geodesics).
Now the idea of Morse is that the flow of the gradient may be used to infer informations about the existence of such critical points. In finite dimension think about a generic function $\mathbb S^2\to \mathbb R$, we know that the sum $n_m - n_s + n_M = 2$ where $n_m, n_s, n_M$ are respectively the number of local minimum, saddles, and local maxima.
Consequently if one knows the number of $n_m, n_s$ he can infer the existence of $n_M$ new critical points.
A famous book about this is Milnor's Morse theory.
-h-principle A PDE on a manifold $M$ prescribes an operator $F:J^r(X)\to \mathbb R$ on some jet bundle. Solutions of the PDE must lie in the zero locus of $F$. $J^{r}(X)$ is fibration over $M$, and a solution of the PDE gives a section of $J^{r}(X)$. Unfortunately a section of $J^r(X)$ does not necessarily have to be induced from a function (such sections are called non-holonomic).
The $h$-principle essentially consists in finding sufficient conditions that ensures that you can promote a section of $J^r(X)$ to an holonomic one (hence a real solution).
A book where you can read about this is Eliashberg & Mishachev "introduction to the h-principle". Also see the introduction of https://arxiv.org/pdf/1609.03180.pdf
-Index theory The Atiyah-Singer index theorem computes the difference between thee dimension of the kernel and the dimension of the cokernel of an elliptic differential operator in purely topological terms.
Studying its proof will give you a lot of tools to understand classic elliptic PDEs and geometry.
A good book about it is Lawson-Michelson's "spin geometry".
