the method of monotone iteration
Nonlinear elliptic equations have no superposition principle and no general solution formula, so the clean linear machinery — Lax-Milgram, the Fredholm alternative — does not apply directly. The method of monotone iteration is a robust constructive way to build a solution anyway, by trapping it between two functions that bracket it from below and above, and then squeezing them together. It is the elliptic cousin of finding a fixed point by starting low and climbing, or starting high and descending.
Consider a semilinear problem - Laplacian u = f(x, u) with Dirichlet data. A subsolution is a function u_low that under-satisfies the equation, - Laplacian u_low is not greater than f(x, u_low); a supersolution u_high over-satisfies it, - Laplacian u_high is not less than f(x, u_high); and they are ordered, u_low is not greater than u_high. Now iterate. Linearise by moving a large constant K across to make the right-hand side monotone (add K u to both sides so f(x, u) + K u is increasing in u), and solve the LINEAR problem - Laplacian u_(n+1) + K u_(n+1) = f(x, u_n) + K u_n repeatedly, starting from u_0 = u_low. Each step is just a linear Dirichlet solve, and the monotone structure plus the maximum principle guarantee the iterates climb monotonically upward and stay below u_high: u_low is not greater than u_1 is not greater than u_2 is not greater than ... is not greater than u_high. A bounded increasing sequence converges, and the limit solves the nonlinear equation. Start instead from u_high and you descend to a (possibly different) solution.
The method is prized for being elementary and almost algorithmic — it needs only the maximum principle, not deep functional analysis — and for delivering existence between the sub- and supersolution, plus a built-in comparison: there is a smallest solution reached from below and a largest reached from above, sandwiching all others. It is a staple for reaction-diffusion steady states, population models, and combustion problems, wherever you can guess a function that is too small and one that is too large. Its main limitation is honest: it relies on having that ordered pair, which requires the nonlinearity to cooperate (typically a monotonicity or a one-sided growth bound), and it finds the extremal solutions, not necessarily every solution in between.
For - Laplacian u = lambda(1 - u) u on a domain with u = 0 on the boundary (a logistic population model), the constant u_high = 1 is a supersolution (since 1 - u = 0 makes the right side zero while -Laplacian of a constant is zero, and the boundary gives slack) and u_low = 0 is a subsolution. Iterating upward from 0 produces an increasing sequence trapped below 1 that converges to a positive steady state — the surviving population — whenever lambda exceeds the principal eigenvalue.
Bracket the solution between an under-shooting subsolution and an over-shooting supersolution, then squeeze.
Monotone iteration finds the minimal and maximal solutions in the bracket, but a nonlinear problem can have additional solutions strictly between them that the method never reaches. And it genuinely depends on finding an ordered sub/supersolution pair — for nonlinearities without the right monotonicity or sign structure, no such pair exists and the method simply does not apply; you must turn to variational or topological (degree-theory) methods instead.