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Solutions to indefinite weakly coupled cooperative elliptic systems

We study the elliptic system \begin{equation*} \begin{cases} -Δu_1 - κ_1u_1 = μ_1|u_1|^{p-2}u_1 + λα|u_1|^{α-2}|u_2|^βu_1, \\ -Δu_2 - κ_2u_2 = μ_2|u_2|^{p-2}u_2 + λβ|u_1|^α|u_2|^{β-2}u_2, \\ u_1,u_2\in D^{1,2}_0(Ω), \end{cases} \end{equation*} where $Ω$ is a bounded domain in $\mathbb{R}^N$, $N\geq 3$, $κ_1,κ_2\in\mathbb{R}$, $μ_1,μ_2,λ>0$, $α,β>1$, and $α+ β= p\le 2^*:=\frac{2N}{N-2}$. For $p\in (2,2^*)$ we establish the existence of a ground state and of a prescribed number of fully nontrivial solutions to this system for $λ$ sufficiently large. If $p=2^*$ and $κ_1,κ_2>0$ we establish the existence of a ground state for $λ$ sufficiently large if, either $N\ge5$, or $N=4$ and neither $κ_1$ nor $κ_2$ are Dirichlet eigenvalues of $-Δ$ in $Ω$.