Let $U\subset\mathbb R^n$ be a bounded Lipschitz domain, let $1<p<\infty$, and let $f:U\times\mathbb R\times\mathbb R^n\to[0,\infty)$ be a Caratheodory integrand, meaning that $x\mapsto f(x,s,\xi)$ is Lebesgue measurable on $U$ for every $(s,\xi)\in\mathbb R\times\mathbb R^n$ and that there exists a set $N\subset U$ with $\mathcal L^n(N)=0$ such that $(s,\xi)\mapsto f(x,s,\xi)$ is continuous from $\mathbb R\times\mathbb R^n$ to $[0,\infty)$ for every $x\in U\setminus N$. Assume that for $\mathcal L^n$-a.e. $x\in U$ and every $s\in\mathbb R$, the map $\xi\mapsto f(x,s,\xi)$ from $\mathbb R^n$ to $[0,\infty)$ is convex. Assume also that there exists a constant $C>0$ such that, for $\mathcal L^n$-a.e. $x\in U$ and every $(s,\xi)\in\mathbb R\times\mathbb R^n$,

\begin{align*} f(x,s,\xi)\le C(1+|s|^p+|\xi|^p). \end{align*}

For each $v\in W^{1,p}(U)$, define

\begin{align*} I[v]:=\int_U f(x,v(x),\nabla v(x))\,d\mathcal L^n(x). \end{align*}

If $(u_k)_{k=1}^{\infty}$ is a sequence in $W^{1,p}(U)$ and $u\in W^{1,p}(U)$ such that

\begin{align*} u_k\rightharpoonup u\quad\text{in }W^{1,p}(U) \end{align*}

and

\begin{align*} u_k\to u\quad\text{in }L^p(U), \end{align*}

then

\begin{align*} I[u]\le \liminf_{k\to\infty} I[u_k]. \end{align*}