= Solution
For $v\in C^0(\overline B)$, let $T(v)$ be the solution from part (i). The weak maximum principle gives
$$
|T(v)|_{0;B}\leq|\varphi|_{0;\partial B}.
$$
On 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 $t\varphi$. Thus the homotopy set is bounded, and the <Leray-Schauder fixed point theorem> supplies $u=T(u)$. By definition,
$$
D_i(a_{ij}(x,u)D_ju)=0,
\qquad u|_{\partial B}=\varphi,
$$
and $u\in W^{1,2}(B)\cap C^0(\overline B)$.
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