There are all sorts of kinds of monotonicity for various PDEs. Generally, one uses monotonicity together with an $\epsilon$-regularity result to get strong restriction on the structure of the singularity set. The most basic monotonicity results take the form $E(f, S_r) \le CE(f, S_R)$ for $E$ some (renormalized) energy of a solution $f$ to some PDE on spheres of radius $r < R$ and $C$ not dependent on $r, R$. A monotonicity of this sort exists for harmonic maps ($d^* d f = 0$) with $E(f,\Omega) = r^{2-n} \|df\|_{L^2(\Omega)}^2$ and Yang-Mills connections $(d_A^* F_A = 0$) with $E(A, \Omega) = r^{4-n} \|F_A\|^2_{L^2(\Omega)}$.