= Solution
For a real solution, $u^4\geq0$, so $\Delta u\geq0$. The <maximum principle for subharmonic functions> gives $u\leq\max_{\partial D}u=0$. The additional nonnegativity condition gives $u\geq0$. Therefore
$$
\boxed{u\equiv0.}
$$
This function satisfies both the equation and boundary condition, so it is exactly the full set of nonnegative solutions. The sign restriction is used alongside the maximum principle, rather than treating the nonlinear equation as a linear uniqueness problem.
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