We study the first eigenvalue of a $\lambda$-dependent cooperative elliptic system involving two quasilinear operators. By variational arguments, we find an expression for the limit of this eigenvalue as $\lambda \rightarrow -\infty$, which improves and extends (for gradient quasilinear systems) a result proved by Álvarez Caudevilla-López Gómez  and Dancer . We apply this result to deduce the existence of strictly principal eigenvalues (i.e., whose eigenfunctions have both components positive) of a weighted system and extend the results proved in Cuesta-Ramos Quoirin  for the scalar case.
"Principal eigenvalue for quasilinear cooperative elliptic systems." Differential Integral Equations 24 (11/12) 1107 - 1124, November/December 2011. https://doi.org/10.57262/die/1356012879