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We prove a Liouville-type theorem for entire solutions of the elliptic system $-\Delta u = |v|^{q-2}v$, $-\Delta v=|u|^{p-2}u$ having finite relative Morse index in the sense of Abbondandolo. Here, $p,q > 2$ and $1/p+1/q> (N-2)/N$. In particular, this yields a result on a priori bounds in $L^{\infty}\times L^{\infty}$ for solutions of superlinear elliptic systems obtained by means of min-max theorems, for both Dirichlet and Neumann boundary conditions.

A priori bounds via the relative Morse index of solutions of an elliptic system