= Solution
We prove existence by the <direct method in the calculus of variations>, making the step with <weakly compact sets> explicit. For $X=W_0^{1,4}(\Omega)$ and $f\in L^{8/7}(\Omega)$, introduce the <functional>
$$
E(u)=\frac14\|\nabla u\|_4^4+\frac18\|u\|_8^8-\int_\Omega fu\,dx.
$$
It is finite by part (b)(i). <Hölder's inequality> and the <Sobolev embedding theorem> give
$$
E(u)\geq\frac14\|u\|_X^4-C_8\|f\|_{8/7}\|u\|_X\longrightarrow+\infty
\quad\text{as }\|u\|_X\to\infty.
$$
Thus $E$ is a <coercive function>, bounded below, and every <minimizing sequence> is bounded in $X$.
The space $X$ is a <reflexive Banach space>: it is isomorphic to a closed <linear subspace> of the finite product of the <reflexive spaces> $L^4(\Omega)$ recording the <function> and its <weak derivatives>. A bounded <sequence> in a <reflexive Banach space> has a <weakly convergent> <subsequence>. Hence, after passing to a <subsequence>, a <minimizing sequence> satisfies $u_j\rightharpoonup u_*$ in $X$. The continuous <linear maps> $u\mapsto\nabla u$ into $L^4$ and $u\mapsto u$ into $L^8$ preserve <weak convergence>, so
$$
\nabla u_j\rightharpoonup\nabla u_*\text{ in }L^4,
\qquad u_j\rightharpoonup u_*\text{ in }L^8.
$$
The <norm> in any <Banach space> has <weak lower semicontinuity>, since it is the supremum of absolute values of bounded <linear functionals> of <dual norm> at most one, by the <Hahn-Banach theorem>. Therefore the two nonnegative <norm> powers in $E$ have <weak lower semicontinuity>. Also $u\mapsto\int_\Omega fu\,dx$ belongs to $X^*$ and is continuous for <weak convergence>. It follows that
$$
E(u_*)\leq\liminf_j E(u_j)=\inf_{u\in X}E(u).
$$
For any $v\in X$, the real <function> $t\mapsto E(u_*+tv)$ has a minimum at $t=0$. Its <derivative> there is
$$
0=\left.\frac{d}{dt}E(u_*+tv)\right|_{t=0}
=\int_\Omega|\nabla u_*|^2\nabla u_*\cdot\nabla v\,dx
+\int_\Omega u_*^7v\,dx-\int_\Omega fv\,dx.
$$
This differentiation is justified directly by finite polynomial expansions: the fourth power of the <Euclidean norm> of $\nabla u_*+t\nabla v$ and the eighth power of $u_*+tv$ have integrable coefficients by <Hölder's inequality>. Thus $A(u_*)=0$, giving existence of a <weak solution>.
If $u,w$ are two <weak solutions>, testing the difference by $u-w$ and using part (b)(iii) gives
$$
0=\langle A(u)-A(w),u-w\rangle\geq\tfrac14\|u-w\|_X^4.
$$
Hence
$$
\boxed{\text{For every }f\in L^{8/7}(\Omega),\text{ there exists exactly one weak solution }u_*\in W_0^{1,4}(\Omega).}
$$
No sign assumption on $f$ or on the <weak solution> is required.
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