Consider the following minimization problem in $\mathbb{R}^N$: $$ \lambda_1(R)：=\min_{(\zeta_i)_{i=1}^m \in H_0^1(B_R(0))} \biggl\{\sum_{i=1}^m \int_{B_R(0)}|\nabla\zeta_i|^2 + \sum_{i,j=1}^m \int_{B_R(0)} H_{ij}\zeta_i\zeta_j \Biggm| \sum_{i=1}^m \int_{B_R(0)} \zeta_i^2 = 1 \biggr\}, $$ where $H_{ij}\in C(\mathbb{R}^N)$ satisfies $$ \sum_{i=1}^m \int_{B_R(0)}|\nabla\zeta_i|^2 + \sum_{i,j=1}^m \int_{B_R(0)} H_{ij}\zeta_i\zeta_j \geq 0 $$ for every $\zeta_k\in C_c^1(B_R(0))$.
It's said in this paper (Page 816) that there exist eigenfunctions $(\zeta_i^R)_{i=1}^m$ such that $$ \Delta\zeta_i^R = \sum_j H_{ij} \zeta_j^R - \lambda_1(R)\zeta_i^R \quad\text{in}\ B_R(0), $$ and $\zeta_i^R = 0$ on $\partial B_R(0)$.
In the classical case, this kind of result can be found in Evans' book, Walter Strauss' book, etc. I want to know how to prove this assertion for system.
Any references about minimization problem and first eigenvalue-eigenfunction of elliptic systems are really appreciated!