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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 $F(p)=\frac{1}{2}|p{|}^2$. The distinct functions $u\equiv 0$ and $v\equiv 1$ 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,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 $n\ge 3$, 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.