= Solution
Take the <functional>
$$
\boxed{J(u)=\int_\Omega\left(\frac14|\nabla u|^4+\frac18|u|^8-fu\right)dx,
\qquad u\in W_0^{1,4}(\Omega).}
$$
The factor $1/8$ is essential: the <derivative> of $|s|^8/8=s^8/8$ is $s^7$, while the <gradient> with respect to $\xi$ of $|\xi|^4/4$ is $|\xi|^2\xi$. As in part (c), finite polynomial expansions and <Hölder's inequality> justify differentiation along every direction $v\in X$, giving $DJ(u)[v]=\langle A(u),v\rangle$.
We prove the global and strict minimum directly. The <strictly convex functions> $\Phi(\xi)=|\xi|^4/4$ and $\Psi(s)=|s|^8/8$ satisfy their tangent inequalities
$$
\Phi(\eta)\geq\Phi(\xi)+|\xi|^2\xi\cdot(\eta-\xi),
\qquad
\Psi(t)\geq\Psi(s)+s^7(t-s),
$$
with strict inequality when the respective arguments differ. For example, strict <convexity> of $\Phi$ follows from strict <operator monotonicity> of its <gradient>, proved by the midpoint identity in part (b)(iii); strict <convexity> of $\Psi$ follows because its <derivative> $s^7$ is strictly increasing.
Apply these inequalities pointwise to $\nabla u_*,\nabla v$ and $u_*,v$, then integrate. The <weak formulation> at $u_*$ with <test function> $v-u_*$ gives
$$
J(v)-J(u_*)\geq\langle A(u_*),v-u_*\rangle=0.
$$
If $v\ne u_*$ in $X$, the <Poincaré inequality> implies $\nabla v\ne\nabla u_*$ on a set of positive <Lebesgue measure>. The first tangent inequality is strict on that set and the second is nonnegative everywhere. Their integrable remainders therefore have strictly positive total <integral>. Hence
$$
\boxed{J(v)>J(u_*)\quad\text{for every }v\in X\setminus\{u_*\}.}
$$
Thus $u_*$ is the unique global <minimizer> of $J$. Conversely, every <minimizer> has zero directional <derivative> in every direction and so satisfies $A(u)=0$, which also identifies it with the unique <weak solution> from part (c).
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