Let $U\subset\mathbb R^n$ be a bounded [open set](/page/Open%20Set), let $1<p<\infty$, and let $\mathcal A\subset W^{1,p}(U)$ be nonempty and sequentially weakly closed in $W^{1,p}(U)$. Let $f:U\times\mathbb R\times\mathbb R^n\to[0,\infty)$ be a Caratheodory integrand. Assume that there exists $C>0$ 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*}

Define the functional $I:\mathcal A\to[0,\infty)$ by

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

Assume that $I$ is coercive on $\mathcal A$, in the sense that $I[u_k]\to\infty$ for every sequence $(u_k)_{k=1}^{\infty}$ in $\mathcal A$ with $\|u_k\|_{W^{1,p}(U)}\to\infty$. Assume also that $I$ is sequentially weakly lower semicontinuous on $\mathcal A$: whenever $u_k,u\in\mathcal A$ and $u_k\rightharpoonup u$ in $W^{1,p}(U)$,

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

Then there exists $u_*\in\mathcal A$ such that

\begin{align*} I[u_*]=\inf_{v\in\mathcal A} I[v]. \end{align*}