Solution
= Solution
\b[False.] On the open unit disk take, for $0<\varepsilon<1$,
$$
\phi(x,y)=-(x^2+y^2)+\varepsilon(x^4+y^4).
$$
Its relevant fourth derivatives satisfy
$$
\phi_{xxxx}+\phi_{yyyy}=48\varepsilon>0.
$$
Nevertheless, since $x^4+y^4\leq(x^2+y^2)^2$, every point with $0<x^2+y^2\leq1$ has
$$
\phi(x,y)\leq-r^2+\varepsilon r^4<0=\phi(0,0).
$$
\b[Thus the maximum lies strictly inside $\Omega$.]