Journal d'Analyse Mathématique

, Volume 137, Issue 1, pp 231–249

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

• Shuangjie Peng
• Yanfang Peng
• Qingfang Wang
Article

## 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.

## References

1. [1]
N. Akhmediev and A. Ankiewicz, Partially coherent solitons on a finite background, Phys. Rev. Lett. 82 (1999), 2661–2664.
2. [2]
A. Ambrosetti and E. Colorado, Standing waves of some coupled nonlinear Schrödinger equations, J. Lond. Math. Soc. 75 (2007), 67–82.
3. [3]
F. Atkinson, H. Brezis, and L. Peletier, Nodal solutions of elliptic equations with the critical Sobolev exponents, J. Differential Equations 85 (1990), 151–170.
4. [4]
T. Bartsch, N. Dancer, and Z.-Q. Wang, A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system, Calc. Var. Partial Differential Equations 37 (2010), 345–361.
5. [5]
T. Bartsch and Z.-Q. Wang, Note on ground states of nonlinear elliptic Schrödinger systems, J. Partial Differential Equations 19 (2006), 200–207.
6. [6]
T. Bartsch, Z.-Q. Wang, and J. Wei, Bounded states for a coupled Schrödinger system, J. Fixed point Theory Appl. 2 (2007), 353–367.
7. [7]
H. Brezis, and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
8. [8]
G. Cerami, S. Solimini, and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal. 69 (1986), 289–306.
9. [9]
Z. Chen, C.-S. Lin, and W. Zou, Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations, J. Differential Equations 255 (2013), 4289–4311.
10. [10]
Z. Chen, C.-S. Lin, and W. Zou, Sign-changing solutions and phase separation for an elliptic system with critical exponent, Comm. Partial Differential Equations 39 (2014), 1827–1859.
11. [11]
Z. Chen, C.-S. Lin, and W. Zou, Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system, Ann. Sc. Norm. Super. Pisa Cl. Sci. 15 (2016), 859–897.
12. [12]
Z. Chen and W. Zou, Positive least energy solutions and phase seperation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal. 205 (2012), 515–551.
13. [13]
Z. Chen and W. Zou, An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calc. Var. Partial Differential Equations 48 (2013), 695–711.
14. [14]
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent:higer dimensional case Calc, Var. Partial Differential Equations 52 (2015), 423–467.
15. [15]
N. Dancer, J. Wei, and T. Weth, A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), 953–969.
16. [16]
G. Devillanova and S. Solimini, A multiplicity result for elliptic equations at critical growth in low dimension, Comm. Contemp. Math. 5 (2003), 171–177.
17. [17]
B. Esry, C. Greene, J. Burke, and J. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett. 78 (1997), 3594–3597.
18. [18]
S. Kim, On vector solutions for coupled nonlinear Schrödinger equations with critical exponents, Comm. Pure Appl. Anal. 12 (2013), 1259–1277.
19. [19]
Y. Kivshar and B. Luther-Davies, Dark optical solitons: physics and applications, Physics Reports 298 (1998), 81–197.
20. [20]
Z. Liu and Z.-Q. Wang, Multiple bound states of nonlinear Schrödinger systems, Comm. Math. Phys. 282 (2008), 721–731.
21. [21]
T. Lin and J. Wei, Ground state of N coupled nonlinear Schrödinger equations in Rn, n ≤ 3, Comm. Math. Phys. 255 (2005), 629–653.
22. [22]
T. Lin and J. Wei, Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), 403–439.
23. [23]
C. Menyuk, Nonlinear pulse propagation in birefringent optical fibers, IEEE J. Quantum Electron. 23 (1987), 174–176.
24. [24]
A. Pomponio, Coupled nonlinear Schrödinger systems with potentials, J. Differential Equations 227 (2006), 258–281.
25. [25]
P. Roselli and M. Willem, Least energy nodal solutions of the Brezis-Nirenberg problem in dimension N = 5, Comm. Contemp. Math. 11 (2009), 59–69.
26. [26]
G. Tarantello, Nodal solutions of semilinear elliptic equations with critical exponent, Differential Integral Equations 5 (1992), 25–42.
27. [27]
J Wei and T. Weth, Asymptotic behavior of solutions of planar elliptic systems with strong competition, Nonlinearity 21 (2008), 305–317.
28. [28]
D. Zhang, On multiple solutions of $$\Delta{u}+\lambda{u}+|u|\frac{4}{n-2}{u}=0$$, Nonlinear Anal. TMA. 13 (1989), 353–372.