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Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. (English) Zbl 1256.35132

Summary: In this paper we study the following coupled Schrödinger system, which can be seen as a critically coupled perturbed Brezis-Nirenberg problem: \[ \begin{cases} -\Delta u + \lambda_1 u = \mu_1 u^3 + \beta uv^2, \quad x \in \Omega, \\ -\Delta v +\lambda_2 v =\mu_2 v^3+\beta vu^2, \quad x\in \Omega, \\ u\geq 0, v\geq 0\, {\mathrm { in }}\, \Omega,\quad u = v = 0 \quad {\text{ on }}\, \partial\Omega.\end{cases} \] Here, \({\Omega \subset \mathbb{R}^4}\) is a smooth bounded domain, \({-\lambda_1(\Omega) < \lambda_1,\lambda_2 < 0 , \mu_1,\mu_2 > 0 }\) and \({\beta \neq 0}\), where \({\lambda_1(\Omega)}\) is the first eigenvalue of \(- \Delta \) with the Dirichlet boundary condition. Note that the nonlinearity and the coupling terms are both critical in dimension 4 (that is, \({\frac{2N}{N-2} = 4}\) when \(N = 4\)). We show that this critical system has a positive least energy solution for negative \(\beta \), positive small \(\beta \) and positive large \(\beta \). For the case in which \({\lambda_1=\lambda_2}\), we obtain the uniqueness of positive least energy solutions. We also study the limit behavior of the least energy solutions in the repulsive case \({\beta\to -\infty}\), and phase separation is expected. These seem to be the first results for this Schrödinger system in the critical case.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B09 Positive solutions to PDEs
35B33 Critical exponents in context of PDEs
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[1] Aubin T.: Problemes isoperimetriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976) · Zbl 0371.46011
[2] Akhmediev N., Ankiewicz A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661–2664 (1999) · doi:10.1103/PhysRevLett.82.2661
[3] Ambrosetti A., Colorado E.: Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 342, 453–458 (2006) · Zbl 1094.35112 · doi:10.1016/j.crma.2006.01.024
[4] Ambrosetti A., Colorado E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. 75, 67–82 (2007) · Zbl 1130.34014 · doi:10.1112/jlms/jdl020
[5] Ambrosotti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[6] Adimurthi , Yadava S.L.: An elementary proof of the uniqueness of positive radial solutions of a quasilinear Dirichlet problem. Arch. Ration. Mech. Anal., 127, 219–229 (1994) · Zbl 0806.35031 · doi:10.1007/BF00381159
[7] Bartsch T., Dancer N., Wang Z.-Q.: A Liouville theorem, a priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. PDE 37, 345–361 (2010) · Zbl 1189.35074 · doi:10.1007/s00526-009-0265-y
[8] Brezis H., Kato T.: Remarks on the Schrodinger operator with singular complex potentials. J. Math. Pures et Appl. 58, 137–151 (1979) · Zbl 0408.35025
[9] Brezis H., Nirenberg L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36, 437–477 (1983) · Zbl 0541.35029 · doi:10.1002/cpa.3160360405
[10] Bartsch T., Wang Z.-Q.: Note on ground states of nonlinear Schrödinger systems. J. Partial Differ. Equ. 19, 200–207 (2006) · Zbl 1104.35048
[11] Bartsch T., Wang Z.-Q., Wang Z.-Q., Wang Z.-Q.: Bound states for a coupled Schrödinger system. J. Fixed Point Theory Appl. 2, 353–367 (2007) · Zbl 1153.35390 · doi:10.1007/s11784-007-0033-6
[12] Byeon, J., Zhang, J., Zou, W.: Singularly perturbed nonlinear Dirichlet problems involving critical growth, preprint. · Zbl 1270.35042
[13] Caffarelli L.A., Lin F.-H.: Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries. J. Am. Math. Soc. 21, 847–862 (2008) · Zbl 1194.35138 · doi:10.1090/S0894-0347-08-00593-6
[14] Caffarelli L.A., Roquejoffre J. M.: Uniform Höder estimates in a class of elliptic systems and applications to singular limits in models for diffusion flames. Arch. Ration. Mech. Anal. 183, 457–487 (2007) · Zbl 1189.35084 · doi:10.1007/s00205-006-0013-9
[15] Conti M., Terracini S., Terracini S., Terracini S.: Asymptotic estimates for the spatial segregation of competitive systems. Adv. Math. 195, 524–560 (2005) · Zbl 1126.35016 · doi:10.1016/j.aim.2004.08.006
[16] Gilbarg, D., Trudinger, N.S.: Elliptic partial Differential Equations of Second Order, 2nd edn. (Grundlehren der mathematischen Wissenschaften, 224). Springer, Berlin, 1983 · Zbl 0562.35001
[17] Dancer N., Wei J., Weth T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 27, 953–969 (2010) · Zbl 1191.35121 · doi:10.1016/j.anihpc.2010.01.009
[18] Esry B., Greene C., Burke J., Bohn J.: Hartree–Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997) · doi:10.1103/PhysRevLett.78.3594
[19] Lieb E., Loss M.: Analysis. American Mathematical Society, Providence (1996)
[20] Lin T., Wei J.: Ground state of N coupled nonlinear Schrödinger equations in $${\(\backslash\)mathbb{R}\^n, n\(\backslash\)geqq 3}$$ . Commun. Math. Phys. 255, 629–653 (2005) · Zbl 1119.35087 · doi:10.1007/s00220-005-1313-x
[21] Lin T., Wei J.: Spikes in two coupled nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 403–439 (2005) · Zbl 1080.35143 · doi:10.1016/j.anihpc.2004.03.004
[22] Lin T., Wei J.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229, 538–569 (2006) · Zbl 1105.35117 · doi:10.1016/j.jde.2005.12.011
[23] Liu Z., Wang Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282, 721–731 (2008) · Zbl 1156.35093 · doi:10.1007/s00220-008-0546-x
[24] Maia L., Montefusco E., Pellacci B.: Positive solutions for a weakly coupled nonlinear Schrödinger systems. J. Differ. Equ. 229, 743–767 (2006) · Zbl 1104.35053 · doi:10.1016/j.jde.2006.07.002
[25] Maia L., Pellacci B., Squassina M.: Semiclassical states for weakly coupled nonlinear Schrödinger systems. J. Eur. Math. Soc. 10, 47–71 (2007) · Zbl 1187.35241
[26] Noris B., Ramos M.: Existence and bounds of positive solutions for a nonlinear Schrödinger system. Proc. Am. Math. Soc. 138, 1681–1692 (2010) · Zbl 1189.35086 · doi:10.1090/S0002-9939-10-10231-7
[27] Noris B., Tavares H., Terracini S., Verzini G.: Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. Commun. Pure Appl. Math. 63, 267–302 (2010) · Zbl 1189.35314
[28] Pomponio A.: Coupled nonlinear Schrödinger systems with potentials. J. Differ. Equ. 227, 258–281 (2006) · Zbl 1100.35098 · doi:10.1016/j.jde.2005.09.002
[29] Sirakov B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in $${\(\backslash\)mathbb{R}\^n}$$ . Commun. Math. Phys. 271, 199–221 (2007) · Zbl 1147.35098 · doi:10.1007/s00220-006-0179-x
[30] Struwe M.: Variational Methods–Applications to Nonlinear Partial Differential Equations and Hamiltonian systems. Springer, Berlin (1996) · Zbl 0864.49001
[31] Talenti G.: Best constant in Sobolev inequality. Ann. Mat. Pure Appl. 110, 352–372 (1976) · Zbl 0353.46018 · doi:10.1007/BF02418013
[32] Wei J., Weth T.: Nonradial symmetric bound states for a system of two coupled Schrödinger equations. Rend. Lincei Mat. Appl. 18, 279–293 (2007) · Zbl 1229.35019
[33] Wei J., Weth T.: Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Ration. Mech. Anal. 190, 83–106 (2008) · Zbl 1161.35051 · doi:10.1007/s00205-008-0121-9
[34] Wei J., Weth T.: Asymptotic behaviour of solutions of planar elliptic systems with strong competition. Nonlinearity 21, 305–317 (2008) · Zbl 1132.35482 · doi:10.1088/0951-7715/21/2/006
[35] Wei, J., Yao, W.: Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Commun. Pure Appl. Anal. (accepted) · Zbl 1264.35237
[36] Willem M.: Minimax Theorems. Birkhäuser, Basel (1996) · Zbl 0856.49001
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