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

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 syste...

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Bibliographic Details
Published inNonlinear analysis: real world applications Vol. 62; p. 103365
Main Authors Huang, Haochuan, Yin, Jingxue, Huang, Rui
Format Journal Article
LanguageEnglish
Published Elsevier Ltd 01.12.2021
Subjects
Lane–Emden system
Periodic heat flow
Sobolev hyperbola
Periodic heat flow
Sobolev hyperbola
Lane–Emden system
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Summary: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.
ISSN:1468-1218
1878-5719
DOI:10.1016/j.nonrwa.2021.103365