Sobolev hyperbola for periodic Lane–Emden heat flow system in N spatial dimension. (December 2021) - Huang, Haochuan; Yin, Jingxue; Huang, Rui
Abstract: The well-known Lane–Emden conjecture indicates that, for the elliptic Lane–Emden system − Δ u = v p, − Δ v = u q in R N, the Sobolev hyperbola 1 p + 1 + 1 q + 1 = N − 2 N 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 u t − Δ u = a ( t ) v p, v t − Δ v = b ( t ) u q in a bounded domain Ω of R N, 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 p q = 1, the nontrivial periodic solutions may exist or not exist.