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Journal d'Analyse Mathématique

, Volume 137, Issue 1, pp 231–249 | Cite as

Sign-changing solutions of an elliptic system with critical exponent in dimension N = 5

  • Shuangjie Peng
  • Yanfang PengEmail author
  • Qingfang Wang
Article
  • 59 Downloads

Abstract

We study the following elliptic system with critical exponent: \(\left\{ {\begin{array}{*{20}{c}} { - \Delta u = {\lambda _1}u + {u_1}|u{|^{2*-2}}u + \beta |u{{|^{\frac{{2*}}{2} - 2}{{u|v|}}^{\frac{{2*}}{2}}}},\;\;x \in \Omega } \\ { - \Delta v = {\lambda _2}v + {u_2}|v{|^{2*-2}}v + \beta |v{{|^{\frac{{2*}}{2} - 2}{{v|u|}}^{\frac{{2*}}{2}}}},\;\;x \in \Omega } \\ \;\;\;\;\;\;\; {u = v = 0,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \in \partial \Omega ,} \end{array}} \right.\;\;\) where Ω is a smooth bounded domain in \(\mathbb{R}^N,\;N=5,2*:=\frac{2N}{N-2}\) is the critical Sobolev exponent, μ1,μ2 > 0, \(\beta \in ( - \sqrt {{\mu _1},{\mu _2}} ,0)\), 0 < λ1, λ2 < λ1(Ω), λ1(Ω) is the first eigenvalue of —Δ in \(H^1_0(\Omega)\). In [10], Chen, Lin and Zou established a sign-changing solution of the above system in the case N ≥ 6 for β < 0 and λ1, λ2 ∈ (0, λ1(Ω)). We show that in dimension N = 5, for λ1 and λ2 slightly smaller than λ1(Ω), the above system has a sign-changing solution in the following sense: one component changes sign and has exactly two nodal domains, while the other one is positive.

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Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral China Normal UniversityWuhan, HubeiChina
  2. 2.School of Mathematical SciencesGiuzhou Normal UniversityGuiyang, GuizhouChina
  3. 3.School of Mathematics and Computer ScienceWuhan Polytechnic UniveristyWuhan, HubeiChina

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