On strongly indefinite systems involving the fractional Laplacian

This paper is devoted to study strongly indefinite systems involving the fractional Laplacian on bounded domains. Explicitly, we obtain existence and non-existence results, apriori estimates of Gidas–Spruck type, and a symmetry result. In addition, we give a different proof for the apriori estimate for nonlinear elliptic problems with the fractional Laplacian obtained in Cabré and Tan (2010) and Tan (2013).