[proofplan] The result is exactly the sequential weak lower semicontinuity theorem of Acerbi and Fusco for nonnegative continuous quasiconvex integrands with upper $p$-growth. We first record the precise external theorem in the form needed here, including the same nonnegativity and growth hypotheses as in the statement. We then verify each hypothesis for the present $\Omega$, $f$, $u_j$, and $u$, and apply the theorem directly to obtain the asserted liminf inequality. [/proofplan]

[step:Record the external Acerbi-Fusco theorem in the needed form]We use the following standard external theorem of Acerbi and Fusco. [claim:Acerbi-Fusco lower semicontinuity theorem] Let $N,M\in\mathbb N$, let $D\subset\mathbb R^N$ be a bounded [open set](/page/Open%20Set), and let $1<q<\infty$. Let \begin{align*} g:\mathbb R^{M\times N}\to[0,\infty) \end{align*} be continuous and quasiconvex in Morrey's sense. Assume that there exists a constant $K>0$ such that, for every $B\in\mathbb R^{M\times N}$, \begin{align*} g(B)\le K(1+|B|^q). \end{align*} If $(v_j)_{j=1}^{\infty}$ is a sequence in $W^{1,q}(D;\mathbb R^M)$, if $v\in W^{1,q}(D;\mathbb R^M)$, and if \begin{align*} v_j\rightharpoonup v\quad\text{in }W^{1,q}(D;\mathbb R^M), \end{align*} then \begin{align*} \int_D g(\nabla v(x))\,d\mathcal L^N(x)\le \liminf_{j\to\infty}\int_D g(\nabla v_j(x))\,d\mathcal L^N(x). \end{align*} [/claim] [proof] This is the classical Acerbi-Fusco sequential lower semicontinuity theorem for integral functionals with quasiconvex integrand and $p$-growth. Its proof is the Acerbi-Fusco blow-up, decomposition, and truncation argument: [weak convergence](/page/Weak%20Convergence) is localized at Lebesgue points of the weak limit, the affine tangent map is subtracted, the rescaled perturbations are replaced by admissible bounded-gradient zero-boundary perturbations without changing the limiting energy, and Morrey quasiconvexity gives the lower density inequality. The nonnegativity of $g$ prevents negative concentration in the limiting energy measures, and the upper $q$-growth supplies integrability and the equiintegrability estimates required by the decomposition lemma. [/proof][/step]

[guided]We need a deep external input, and the exact input must match the hypotheses of the theorem being proved. The standard Acerbi-Fusco theorem says the following. Suppose $D\subset\mathbb R^N$ is bounded and open, $1<q<\infty$, and \begin{align*} g:\mathbb R^{M\times N}\to[0,\infty) \end{align*} is continuous, Morrey quasiconvex, and satisfies an upper $q$-growth estimate \begin{align*} g(B)\le K(1+|B|^q) \end{align*} for all $B\in\mathbb R^{M\times N}$, where $K>0$ is fixed. Then the functional \begin{align*} v\mapsto \int_D g(\nabla v(x))\,d\mathcal L^N(x) \end{align*} is sequentially weakly lower semicontinuous on $W^{1,q}(D;\mathbb R^M)$: whenever $v_j\rightharpoonup v$ in $W^{1,q}(D;\mathbb R^M)$, \begin{align*} \int_D g(\nabla v(x))\,d\mathcal L^N(x)\le \liminf_{j\to\infty}\int_D g(\nabla v_j(x))\,d\mathcal L^N(x). \end{align*} Why is this the right external theorem rather than a shorter quasiconvexity argument? Quasiconvexity applies to bounded zero-boundary perturbations on a fixed domain, while weak convergence in $W^{1,q}$ produces only weakly convergent gradients and may create concentration. The Acerbi-Fusco decomposition and truncation argument is precisely the theorem that bridges this gap: after blow-up at almost every point, it replaces the weak perturbations by admissible bounded-gradient perturbations while preserving the limiting lower energy bound. The assumption $g\ge 0$ is also part of the theorem as stated here; it rules out negative concentration when the energy densities are passed to limiting Radon measures.[/guided]

[step:Verify the hypotheses of the external theorem]Apply the external theorem with \begin{align*} N:=n, \end{align*} with \begin{align*} M:=m, \end{align*} with \begin{align*} D:=\Omega, \end{align*} with \begin{align*} q:=p, \end{align*} and with \begin{align*} g:=f. \end{align*} The set $\Omega$ is bounded and open by hypothesis. The exponent satisfies $1<p<\infty$ by hypothesis. The map $f:\mathbb R^{m\times n}\to[0,\infty)$ is continuous by hypothesis, and its Morrey quasiconvexity is exactly the quasiconvexity condition required in the external theorem. The stated upper $p$-growth condition gives the required growth estimate with the same constant $K:=C$. The sequence $(u_j)_{j=1}^{\infty}$ belongs to $W^{1,p}(\Omega;\mathbb R^m)$, the map $u$ belongs to $W^{1,p}(\Omega;\mathbb R^m)$, and \begin{align*} u_j\rightharpoonup u\quad\text{in }W^{1,p}(\Omega;\mathbb R^m) \end{align*} by hypothesis. Thus every hypothesis of the Acerbi-Fusco lower semicontinuity theorem is satisfied.[/step]

[guided]We now match the abstract theorem to the present notation. The external theorem uses the symbols $N$, $M$, $D$, $q$, and $g$. In the present theorem these are \begin{align*} N:=n, \end{align*} \begin{align*} M:=m, \end{align*} \begin{align*} D:=\Omega, \end{align*} \begin{align*} q:=p, \end{align*} and \begin{align*} g:=f. \end{align*} We check the hypotheses one by one. First, the external theorem requires $D$ to be a bounded open subset of $\mathbb R^N$. This is satisfied because the theorem statement assumes $\Omega\subset\mathbb R^n$ is bounded and open. Second, the external theorem requires $1<q<\infty$. This is satisfied because the theorem statement assumes $1<p<\infty$. Third, the external theorem requires \begin{align*} g:\mathbb R^{M\times N}\to[0,\infty) \end{align*} to be continuous and Morrey quasiconvex. After the substitution $g=f$, $M=m$, and $N=n$, this is exactly the stated assumption that \begin{align*} f:\mathbb R^{m\times n}\to[0,\infty) \end{align*} is continuous and quasiconvex in Morrey's sense. Fourth, the external theorem requires a constant $K>0$ such that \begin{align*} g(B)\le K(1+|B|^q) \end{align*} for all $B\in\mathbb R^{M\times N}$. The present growth hypothesis gives this with $K:=C$, $q:=p$, and $B:=A$. Finally, the external theorem requires weak convergence in the [Sobolev space](/page/Sobolev%20Space) $W^{1,q}(D;\mathbb R^M)$. Under the same substitutions, this is the hypothesis \begin{align*} u_j\rightharpoonup u\quad\text{in }W^{1,p}(\Omega;\mathbb R^m). \end{align*} Therefore the external theorem applies without changing any hypothesis or adding any hidden assumption.[/guided]

[step:Apply Acerbi-Fusco lower semicontinuity to the given sequence] By the Acerbi-Fusco lower semicontinuity theorem applied with the identifications verified above, \begin{align*} \int_\Omega f(\nabla u(x))\,d\mathcal L^n(x)\le \liminf_{j\to\infty}\int_\Omega f(\nabla u_j(x))\,d\mathcal L^n(x). \end{align*} This is precisely the asserted weak lower semicontinuity inequality. [/step]