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We study the elliptic system \begin{equation*} \begin{cases} -\Delta u_1 -\kappa_1u_1 = \mu_1|u_1|^{p-2}u_1 + \lambda\alpha|u_1|^{\alpha-2}|u_2|^\betau_1, \\ -\Delta u_2 - \kappa_2u_2 = \mu_2|u_2|^{p-2}u_2 +\lambda\beta|u_1|^\alpha|u_2|^{\beta-2}u_2, \\ u_1,u_2\in D^{1,2}_0(\Omega),\end{cases} \end{equation*} where $\Omega$ is a bounded domain in$\mathbb{R}^N$, $N\geq 3$, $\kappa_1,\kappa_2\in\mathbb{R}$,$\mu_1,\mu_2,\lambda>0$, $\alpha,\beta>1$, and $\alpha + \beta = p\le2^*:=\frac{2N}{N-2}$.  For $p\in (2,2^*)$ we establish the existence of a ground state and of aprescribed number of fully nontrivial solutions to this system for $\lambda$sufficiently large.  If $p=2^*$ and $\kappa_1,\kappa_2>0$ we establish the existence of a groundstate for $\lambda$ sufficiently large if, either $N\ge5$, or $N=4$ and neither$\kappa_1$ nor $\kappa_2$ are Dirichlet eigenvalues of $-\Delta$ in $\Omega$.

Solutions to indefinite weakly coupled cooperative elliptic systems