= Solution
The boundary data are encoded by the <Affine Sobolev space>
$$
\mathcal A_\varphi=\varphi+W_0^{1,p}(\Omega)
=\{u\in W^{1,p}(\Omega):\operatorname{Tr}u=\operatorname{Tr}\varphi\}.
$$
A function $u\in\mathcal A_\varphi$ is a weak solution of the homogeneous <p-Laplacian equation> when
$$
\int_\Omega|Du|^{p-2}Du\mathbin\cdot D\psi=0
\qquad\text{for every }\psi\in W_0^{1,p}(\Omega).
$$
For existence, take a minimizing sequence for the <p-energy> on $\mathcal A_\varphi$. The <Poincare inequality> bounds $u-\varphi$ in $W^{1,p}$ by its gradient, so the sequence is bounded in the <reflexive Banach space> $W^{1,p}$. A weakly convergent subsequence remains in the weakly closed affine space, and convexity of $|\eta|^p$ gives <weak lower semicontinuity>. The <direct method in the calculus of variations> therefore produces a minimizer, whose first variation is precisely the displayed weak equation.
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