= Solution
The missing hypothesis is \b[<uniform ellipticity> up to the boundary]. The principal coefficient matrix and its <quadratic form> are
$$
a(x,y)=\begin{pmatrix}1&0\\0&y^2\end{pmatrix},\qquad a^{ij}\xi_i\xi_j=\xi_1^2+y^2\xi_2^2.
$$
Although this is <positive-definite> at every interior point, no single positive <uniform ellipticity> constant works on $R$: taking $\xi=(0,1)$ requires $y^2\geq\lambda$ for arbitrarily small positive $y$. The normal second-derivative coefficient vanishes on the lower side. This <degenerate ellipticity> allows the equation and boundedness to force a boundary trace, instead of allowing arbitrary continuous <Dirichlet data>. The coefficient matrix is <positive-definite> at every interior point, but lacks the uniform lower bound used by the usual bounded-domain existence theorem.
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