We study the weakly coupled critical elliptic system
$ \begin{equation*} \begin{cases} -\Delta u = \mu_{1}|u|^{2^{*}-2}u+\lambda\alpha |u|^{\alpha-2}|v|^{\beta}u & \text{in }\Omega,\\ -\Delta v = \mu_{2}|v|^{2^{*}-2}v+\lambda\beta |u|^{\alpha}|v|^{\beta-2}v & \text{in }\Omega,\\ u = v = 0 & \text{on }\partial\Omega, \end{cases} \end{equation*} $
where $ \Omega $ is a bounded smooth domain in $ \mathbb{R}^{N} $, $ N\geq 3 $, $ 2^{*}: = \frac{2N}{N-2} $ is the critical Sobolev exponent, $ \mu_{1},\mu_{2}>0 $, $ \alpha, \beta>1 $, $ \alpha+\beta = 2^{*} $ and $ \lambda\in\mathbb{R} $.
We establish the existence of a prescribed number of fully nontrivial solutions to this system under suitable symmetry assumptions on $ \Omega $, which allow domains with finite symmetries, and we show that the positive least energy symmetric solution exhibits phase separation as $ \lambda\to -\infty $.
We also obtain existence of infinitely many solutions to this system in $ \Omega = \mathbb{R}^N $.
Citation: |
[1] |
A. Bahri and J.-M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math., 41 (1988), 253-294.
doi: 10.1002/cpa.3160410302.![]() ![]() ![]() |
[2] |
A. Castro, J. Cossio and J. M. Neuberger, A sign-changing solution for a superlinear Dirichlet problem, Rocky Mountain J. Math., 27 (1997), 1041-1053.
doi: 10.1216/rmjm/1181071858.![]() ![]() ![]() |
[3] |
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent, Arch. Ration. Mech. Anal., 205 (2012), 515-551.
doi: 10.1007/s00205-012-0513-8.![]() ![]() ![]() |
[4] |
Z. Chen and W. Zou, Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: Higher dimensional case, Calc. Var. Partial Differential Equations, 52 (2015), 423-467.
doi: 10.1007/s00526-014-0717-x.![]() ![]() ![]() |
[5] |
M. Clapp, Entire nodal solutions to the pure critical exponent problem arising from concentration, J. Differential Equations, 261 (2016), 3042-3060.
doi: 10.1016/j.jde.2016.05.013.![]() ![]() ![]() |
[6] |
M. Clapp and J. Faya, Multiple solutions to the Bahri-Coron problem in some domains with nontrivial topology, Proc. Amer. Math. Soc., 141 (2013), 4339-4344.
doi: 10.1090/S0002-9939-2013-12043-5.![]() ![]() ![]() |
[7] |
M. Clapp and J. Faya, Multiple solutions to anisotropic critical and supercritical problems in symmetric domains, in Contributions to Nonlinear Elliptic Equations and Systems, 99-120, Progr. Nonlinear Differential Equations Appl., 86, Birkhäuser/Springer, Cham, 2015.
doi: 10.1007/978-3-319-19902-3_8.![]() ![]() ![]() |
[8] |
M. Clapp and F. Pacella, Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size, Math. Z., 259 (2008), 575-589.
doi: 10.1007/s00209-007-0238-9.![]() ![]() ![]() |
[9] |
M. Clapp and A. Pistoia, Existence and phase separation of entire solutions to a pure critical competitive elliptic system, Calc. Var. Partial Differential Equations, 57 (2018), Art. 23, 20 pp.
doi: 10.1007/s00526-017-1283-9.![]() ![]() ![]() |
[10] |
M. Conti, S. Terracini and G. Verzini, Nehari's problem and competing species systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 871-888.
doi: 10.1016/S0294-1449(02)00104-X.![]() ![]() ![]() |
[11] |
M. Conti, S. Terracini and G. Verzini, Asymptotic estimates for the spatial segregation of competitive systems, Adv. Math., 195 (2005), 524-560.
doi: 10.1016/j.aim.2004.08.006.![]() ![]() ![]() |
[12] |
M. del Pino, M. Musso, F. Pacard and A. Pistoia, Large energy entire solutions for the Yamabe equation, J. Differential Equations, 251 (2011), 2568-2597.
doi: 10.1016/j.jde.2011.03.008.![]() ![]() ![]() |
[13] |
W. Ding, On a conformally invariant elliptic equation on $\mathbb{R}^n$, Comm. Math. Phys., 107 (1986), 331-335.
doi: 10.1007/BF01209398.![]() ![]() ![]() |
[14] |
B. D. Esry, C. H. Greene, J. P. Burke Jr. and J. L. Bohn, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78 (1997), 3594-3597.
doi: 10.1103/PhysRevLett.78.3594.![]() ![]() |
[15] |
Y. Ge, M. Musso and A. Pistoia, Sign changing tower of bubbles for an elliptic problem at the critical exponent in pierced non-symmetric domains, Comm. Partial Differential Equations, 35 (2010), 1419-1457.
doi: 10.1080/03605302.2010.490286.![]() ![]() ![]() |
[16] |
Y. Guo, B. Li and J. Wei, Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in $\mathbb{R}^3$, J. Differential Equations, 256 (2014), 3463-3495.
doi: 10.1016/j.jde.2014.02.007.![]() ![]() ![]() |
[17] |
J. Liu, X. Liu and Z.-Q. Wang, Sign-changing solutions for coupled nonlinear Schrödinger equations with critical growth, J. Differential Equations, 261 (2016), 7194-7236.
doi: 10.1016/j.jde.2016.09.018.![]() ![]() ![]() |
[18] |
S. Peng, Y.-F. Peng and Z.-Q. Wang, On elliptic systems with Sobolev critical growth, Calc. Var. Partial Differential Equations, 55 (2016), Art. 142, 30 pp.
doi: 10.1007/s00526-016-1091-7.![]() ![]() ![]() |
[19] |
A. Pistoia and N. Soave, On Coron's problem for weakly coupled elliptic systems, Proc. Lond. Math. Soc., 116 (2018), 33-67.
doi: 10.1112/plms.12073.![]() ![]() ![]() |
[20] |
A. Pistoia and H. Tavares, Spiked solutions for Schrödinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions, J. Fixed Point Theory Appl., 19 (2017), 407-446.
doi: 10.1007/s11784-016-0360-6.![]() ![]() ![]() |
[21] |
N. Soave, On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differential Equations, 53 (2015), 689-718.
doi: 10.1007/s00526-014-0764-3.![]() ![]() ![]() |
[22] |
A. Szulkin, Ljusternik–Schnirelmann theory on $\mathcal{C}^1$-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119-139.
doi: 10.1016/S0294-1449(16)30348-1.![]() ![]() ![]() |
[23] |
E. Timmermans, Phase separation of Bose-Einstein condensates, Phys. Rev. Lett., 81 (1998), 5718-5721.
doi: 10.1103/PhysRevLett.81.5718.![]() ![]() |
[24] |
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, 24. Birkhäuser Boston, Inc., Boston, MA, 1996.
doi: 10.1007/978-1-4612-4146-1.![]() ![]() ![]() |