= Solution
For the spaces in part (b)(i) and fixed $f\in Y$, define $A:X\to X^*$ by
$$
\boxed{\langle A(u),v\rangle
=\int_\Omega|\nabla u|^2\nabla u\cdot\nabla v\,dx
+\int_\Omega u^7v\,dx-\int_\Omega fv\,dx.}
$$
For each $u$, this is a <linear functional> in $v$, and the estimates in part (b)(i) show
$$
\|A(u)\|_{X^*}\leq\|u\|_X^3+C_8^8\|u\|_X^7+C_8\|f\|_{8/7}.
$$
Hence $A(u)$ belongs to the <continuous dual space> $X^*$, and $A$ sends bounded <sets> to bounded <sets>. The <weak formulation> is exactly
$$
\boxed{\langle A(u),v\rangle=0\quad\forall v\in X,\quad\text{equivalently }A(u)=0\text{ in }X^*.}
$$
The forcing term is included in $A$ so that its right side is zero, as requested.
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