# American Institute of Mathematical Sciences

June  2019, 39(6): 3265-3289. doi: 10.3934/dcds.2019135

## Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains

 Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, Ciudad Universitaria, 04510 Coyoacán, CDMX, Mexico

Received  June 2018 Published  February 2019

Fund Project: Mónica Clapp was partially supported by CONACYT grant 237661 (Mexico) and UNAM-DGAPA-PAPIIT grant IN100718 (Mexico). Jorge Faya was supported by a postdoctoral fellowship under CONACYT grant 237661 (Mexico).

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: Mónica Clapp, Jorge Faya. Multiple solutions to a weakly coupled purely critical elliptic system in bounded domains. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3265-3289. doi: 10.3934/dcds.2019135
##### References:
 [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.

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##### References:
 [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.
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