= Solution
\b[No: the conclusion need not survive without an additional restriction at infinity.] A counterexample that keeps the <functional> finite is $\Omega=\mathbb R^n$ and $F(p)=\tfrac12|p|^2$. The distinct functions $u\equiv0$ and $v\equiv1$ both lie in $C^2(\overline\Omega)$ and have energy zero. Their boundary data agree because $\partial\Omega$ is empty. For any variation with finite <gradient> energy,
$$
\mathcal F(u+t\varphi)=\mathcal F(v+t\varphi)=\frac{t^2}{2}\int_{\mathbb R^n}|D\varphi|^2,
$$
so both are genuine <critical points>, not merely formal solutions with infinite <functional> value.
With nonempty boundary, the classical <Euler-Lagrange equation> also illustrates the missing condition: in dimension $n\geq3$, on $\Omega=\{|x|>1\}$, both $0$ and $1-|x|^{2-n}$ are <harmonic functions> with zero boundary values. The latter has finite <Dirichlet energy>, because its gradient-energy integral is proportional to $\int_1^\infty r^{1-n}dr$, and is stationary under <compactly supported> variations. It approaches $1$ at infinity while the zero solution approaches $0$.
This is <nonuniqueness of Dirichlet data without control at infinity>. For a global variational formulation one must specify the admissible variations: the exterior example is not critical against every finite-energy zero-boundary variation, since varying it in its own direction changes its energy to first order. Prescribing an appropriate limit at infinity, or a variational space in which the difference is an admissible zero-boundary <test function> and constants are excluded, can restore uniqueness.
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