Liouville-type theorem of positive periodic solutions for the periodic parabolic system

Abstract

ABSTRACT In this paper, we are concerned with the problem for a periodic parabolic system where p, q>0, Ω is a bounded and smooth domain in , and are properly smooth, bounded and positive periodic functions with periodicity ω. It is known that for the elliptic system in , there exists a Sobolev hyperbola , where is defined by (3), which characterizes the existence and nonexistence of nontrivial nonnegative solutions. By using the classical blowing-up method, Liouville-type theorem, Leray–Schauder fixed point theory and Pohozaev identity, we will prove that the Sobolev hyperbola is also a critical curve for the existence and non-existence of the positive periodic solutions for the periodic parabolic system.