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

DOI: http://dx.doi.org/10.12775/TMNA.2009.027

#### Abstract

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.

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.

#### Keywords

Elliptic system; strongly indefinite functional; Morse index; Lyapunov-Schmidt reduction

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