Let $\Omega\subset \mathbb R^n$ be a bounded Lipschitz domain, let $p>1$ with $n\le p<\infty$, and let $\Gamma\subset\partial\Omega$ be a relatively open boundary part with $\mathcal H^{n-1}(\Gamma)>0$. Let $g\in W^{1,p}(\Omega;\mathbb R^n)$ be prescribed trace data. Let $\mathcal C\subset W^{1,p}(\Omega;\mathbb R^n)$ be a sequentially weakly closed constraint class encoding the finite-energy determinant positivity condition and the imposed non-interpenetration condition. Define the admissible class by $\mathcal A:=\{u\in\mathcal C:\operatorname{Tr}u=\operatorname{Tr}g\text{ on }\Gamma\}$. Assume $\mathcal A$ is nonempty and sequentially weakly closed in $W^{1,p}(\Omega;\mathbb R^n)$.

For $1\le r\le n$, let $N_r$ be the number of $r\times r$ minors of an $n\times n$ matrix, and let $M_r:\mathbb R^{n\times n}\to \mathbb R^{N_r}$ denote the map sending a matrix to the vector of all its $r\times r$ minors, with $M_1(F)=F$. Let $N=\sum_{r=1}^n N_r$ and define $M:\mathbb R^{n\times n}\to \mathbb R^N$ by $M(F)=(M_1(F),\dots,M_n(F))$.

Let $W:\Omega\times \mathbb R^n\times \mathbb R^{n\times n}\to (-\infty,\infty]$ be a stored-energy density. Assume that there is a normal integrand $G:\Omega\times \mathbb R^n\times \mathbb R^N\to (-\infty,\infty]$ such that, for $\mathcal L^n$-a.e. $x\in\Omega$ and every $(y,F)\in\mathbb R^n\times\mathbb R^{n\times n}$, $W(x,y,F)=G(x,y,M(F))$. Assume that, for $\mathcal L^n$-a.e. $x\in\Omega$, the map $(y,Z)\mapsto G(x,y,Z)$ is lower semicontinuous and, for every $y\in\mathbb R^n$, the map $Z\mapsto G(x,y,Z)$ is convex.

Assume that there are exponents $q_2,\dots,q_n>1$ and constants $c>0$ and $C\ge 0$ such that, for $\mathcal L^n$-a.e. $x\in\Omega$ and every $(y,F)\in\mathbb R^n\times\mathbb R^{n\times n}$, $W(x,y,F)\ge c\left(|F|^p+\sum_{r=2}^{n}|M_r(F)|^{q_r}\right)-C$. Assume also that $W(x,y,F)=+\infty$ whenever $\det F\le 0$, and that $W(x,y,F)\to+\infty$ as $\det F\downarrow 0$ locally uniformly for bounded $(x,y,F)$ with $\det F>0$.

Let $\ell:\mathcal A\to\mathbb R$ be sequentially weakly continuous and satisfy the subcritical loading estimate: there exist constants $a,b\ge 0$ and an exponent $0\le \theta<p$ such that $|\ell[u]|\le a\|u\|_{W^{1,p}(\Omega)}^\theta+b$ for every $u\in\mathcal A$. Define $I:\mathcal A\to(-\infty,\infty]$ by $I[u]=\int_\Omega W(x,u(x),\nabla u(x))\,d\mathcal L^n(x)-\ell[u]$. Then there exists $u_*\in\mathcal A$ such that $I[u_*]=\inf_{u\in\mathcal A}I[u]$.