Let $U\subset\mathbb R^n$ be a bounded Lipschitz domain, let $1<p<\infty$, and let $g\in W^{1,p}(U)$. Define the affine zero-trace class

\begin{align*} g+W^{1,p}_0(U)=\{g+w:w\in W^{1,p}_0(U)\}. \end{align*}

Let $\mathcal A\subset g+W^{1,p}_0(U)$ 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, meaning that $x\mapsto f(x,s,\xi)$ is measurable for every $(s,\xi)\in\mathbb R\times\mathbb R^n$ and $(s,\xi)\mapsto f(x,s,\xi)$ is continuous for $\mathcal L^n$-a.e. $x\in U$. Assume that:

1. for $\mathcal L^n$-a.e. $x\in U$ and every $s\in\mathbb R$, the map $\xi\mapsto f(x,s,\xi)$ is convex on $\mathbb R^n$; 2. there exist $\alpha>0$ and $a\in L^1(U)$ such that, for $\mathcal L^n$-a.e. $x\in U$ and all $(s,\xi)\in\mathbb R\times\mathbb R^n$,

\begin{align*} f(x,s,\xi)\ge \alpha |\xi|^p-a(x); \end{align*}

3. there exists $C>0$ such that, for $\mathcal L^n$-a.e. $x\in U$ and all $(s,\xi)\in\mathbb R\times\mathbb R^n$,

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

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

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

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

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