Uniform a priori estimates for positive solutions of the Lane–Emden system in the plane

Abstract

We prove that positive solutions of the superlinear Lane–Emden system in a two-dimensional smooth bounded domain are bounded independently of the exponents in the system, provided the exponents are comparable. As a consequence, the energy of the solutions is uniformly bounded, a crucial information in their asymptotic study. In addition, the boundedness may fail if the exponents are not comparable.