Sobolev hyperbola for periodic Lane–Emden heat flow system in N spatial dimension

Abstract

The well-known Lane–Emden conjecture indicates that, for the elliptic Lane–Emden system −Δu=vp, −Δv=uq in RN, the Sobolev hyperbola 1p+1+1q+1=N−2N is expected as the critical curve for the existence and nonexistence of entire solutions. In this paper, we study the periodic Lane–Emden heat flow system ut−Δu=a(t)vp, vt−Δv=b(t)uq in a bounded domain Ω of RN, subject to homogeneous Dirichlet boundary condition. We will show that the Sobolev hyperbola is also a critical curve for the existence and nonexistence of periodic solutions. Moreover, if pq=1, the nontrivial periodic solutions may exist or not exist.