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  "body": "\n## Summary\n\nThe Leray–Schauder method proves that a nonlinear equation $u = T(u)$ with a compact map $T$ has a solution.\nIt deforms the equation into one that is easy to solve.\nOne embeds the problem in a family $u = T(\\lambda, u)$, $0 \\le \\lambda \\le 1$, and proves an a priori bound: all solutions for all $\\lambda$ lie in a fixed ball.\nThe [[Leray–Schauder degree|3VJC5Q9MYA]] is then the same for every $\\lambda$, and a nonzero degree at $\\lambda = 0$ forces a solution at $\\lambda = 1$.\nIt is a main tool for nonlinear integral equations and for quasilinear elliptic boundary value problems.\n\n## Explanations\n\n### L1 human\n\nImagine slowly turning a knob that changes an easy equation into the hard one you care about.\nSolutions move as the knob turns, but they cannot vanish into thin air unless they escape to infinity.\nIf one can show that no solution ever gets too large, then the solution of the easy equation must survive until the end.\nThe degree is the count of solutions, with signs, that stays fixed along the way.\n",
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