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For $v\in C^0(\overline{B})$, let $T(v)$ be the solution from part (i). The weak maximum principle givesOn this fixed bounded range of values, $\lambda$ and $\Lambda$ have common positive lower and finite upper bounds. The global De Giorgi-Nash-Moser theorem therefore bounds $T(v)$ uniformly in one Hölder space. The compact embedding $C^{0,\mu }(\overline{B})\hookrightarrow C^0(\overline{B})$ makes $T$ compact. Uniform convergence $v_k\to v$, the continuity of $a_{ij}$ on the relevant compact set, energy bounds, and uniqueness of the limiting linear problem show that $T(v_k)\to T(v)$ uniformly, so $T$ is continuous.

The same maximum-principle bound controls every solution of the Leray–Schauder homotopy after using boundary data $t\phi$. Thus the homotopy set is bounded, and the Leray-Schauder fixed point theorem supplies $u=T(u)$. By definition,and $u\in W^{1,2}(B)\cap C^0(\overline{B})$.

The Leray-Schauder fixed point theorem says that if $X$ is a Banach space and $T:X\to X$ is continuous and compact, and the setis bounded, then $T$ has a fixed point.

The Leray-Schauder fixed point theorem says that a continuous compact map $T$ on a Banach space has a fixed point if the homotopy setis bounded. If $u=tT(u)$ with $t>0$, multiplying the equation for $T(u)$ by $t$ shows that $u$ solvesThe case $t=0$ gives $u=0$. Therefore it is enough to prove one uniform $C^{1,\beta }$ estimate for all solutions of this family and all $t\in [0,1]$. The theorem then yields a fixed point $u=T(u)$, which solves the original quasilinear problem.

Initially the construction gives $u\in C^{2,\alpha \beta }$. This makes $u$ and $Du$ Lipschitz, so composing the original $C^{0,\alpha }$ coefficients with the jet of $u$ produces $C^{0,\alpha }$ coefficients. A second application of the global Schauder estimate gives $u\in C^{2,\alpha }(\overline{\Omega })$. The fixed-point theorem is an existence result and supplies no uniqueness.