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

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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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Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei, 430079, China

    Shuangjie Peng

  2. School of Mathematical Sciences, Giuzhou Normal University, Guiyang, Guizhou, 550001, China

    Yanfang Peng

  3. School of Mathematics and Computer Science, Wuhan Polytechnic Univeristy, Wuhan, Hubei, 430079, China

    Qingfang Wang

Authors
  1. Shuangjie Peng
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  2. Yanfang Peng
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  3. Qingfang Wang
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Correspondence to Yanfang Peng.

Additional information

Partially supported by NSFC (No. 11831009, No. 11571130)

Partially supported by NSFC (No. 11501143) and the Ph.D. launch scientific research projects of Guizhou Normal University (No. 2014).

Supported by NSFC (N0. 11701439)

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Peng, S., Peng, Y. & Wang, Q. Sign-changing solutions of an elliptic system with critical exponent in dimension N = 5. JAMA 137, 231–249 (2019). https://doi.org/10.1007/s11854-018-0071-6

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  • DOI: https://doi.org/10.1007/s11854-018-0071-6

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