Sign changing solutions for coupled critical elliptic equations. (English) Zbl 1437.35277
Summary: In this paper, we consider the coupled elliptic system with a Sobolev critical exponent. We show the existence of a sign changing solution for problem \(\left(\mathscr{P}\right)\) for the coupling parameter \(-\sqrt{\mu_1 \mu_2} < \beta < 0\). We also construct multiple sign changing solutions for the symmetric case.
MSC:
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
References:
| [1] | Akhmediev, N.; Ankiewicz, A., Partially coherent solitons on a finite background, Physical Review Letters, 82, 13, 2661-2664 (1999) · doi:10.1103/physrevlett.82.2661 |
| [2] | Sirakov, B., Least energy solitary waves for a system of a nonlinear Schrödinger equations in \(\mathbb{R}^n\), Communications in Mathematical Physics, 271, 1, 199-221 (2007) · Zbl 1147.35098 · doi:10.1007/s00220-006-0179-x |
| [3] | Esry, B. D.; Greene, C. H.; Burke, J. P.; Bohn, J. L., Hartree-fock theory for double condesates, Physical Review Letters, 78, 19, 3594-3597 (1997) · doi:10.1103/physrevlett.78.3594 |
| [4] | Ambrosetti, A.; Colorado, E., Standing waves of some coupled nonlinear Schrödinger equations, Journal of the London Mathematical Society, 75, 1, 67-82 (2007) · Zbl 1130.34014 · doi:10.1112/jlms/jdl020 |
| [5] | Bartsch, T.; Dancer, N.; Wang, Z.-Q., A Liouville theorem, a priori bounds and bifurcating branches of positive solutions for a nonlinear elliptic system, Calculus of Variations and Partial Differential Equations, 37, 3-4, 345-361 (2010) · Zbl 1189.35074 · doi:10.1007/s00526-009-0265-y |
| [6] | Bartsch, T.; Wang, Z.-Q., Note on ground states of nonlinear Schrödinger systems, Journal of Partial Differential Equations, 19, 200-207 (2006) · Zbl 1104.35048 |
| [7] | Bartsch, T.; Wang, Z.-Q.; Wei, J., Bound states for a coupled Schrödinger system, Journal of Fixed Point Theory and Applications, 2, 2, 353-367 (2007) · Zbl 1153.35390 · doi:10.1007/s11784-007-0033-6 |
| [8] | Chen, Z.; Lin, C.-S.; Zou, W., Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system, Annali della Scuola Normale Superiore di Pisa, 15, 859-897 (2016) · Zbl 1343.35101 |
| [9] | Chen, Z.; Lin, C.-S.; Zou, W., Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations, Journal of Differential Equations, 255, 11, 4289-4311 (2013) · Zbl 1281.35073 · doi:10.1016/j.jde.2013.08.009 |
| [10] | Chen, Z.; Zou, W., An optimal constant for the existence of least energy solutions of a coupled Schrödinger system, Calculus of Variations and Partial Differential, 48, 3-4, 695-711 (2013) · Zbl 1286.35104 · doi:10.1007/s00526-012-0568-2 |
| [11] | Conti, M.; Terracini, S.; Verzini, G., Asymptotic estimates for the spatial segregation of competitive systems, Advances in Mathematics, 195, 2, 524-560 (2005) · Zbl 1126.35016 · doi:10.1016/j.aim.2004.08.006 |
| [12] | Conti, M.; Terracini, S.; Verzini, G., Nehari’s problem and competing species systems, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 19, 6, 871-888 (2002) · Zbl 1090.35076 · doi:10.1016/s0294-1449(02)00104-x |
| [13] | Dancer, E. N.; Wei, J.-C.; Weth, T., A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 27, 3, 953-969 (2010) · Zbl 1191.35121 · doi:10.1016/j.anihpc.2010.01.009 |
| [14] | Lingua, T.-C.; Wei, J., Ground state of N-coupled nonlinear Schrödinger equations in \(\mathbb{R}^N, N\leq3\), Communications in Mathematical Physics, 255, 3, 629-653 (2005) · Zbl 1119.35087 · doi:10.1007/s00220-005-1313-x |
| [15] | Lin, T.-C.; Wei, J.-C., Spikes in two coupled nonlinear Schrödinger equations, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 22, 4, 403-439 (2005) · Zbl 1080.35143 · doi:10.1016/j.anihpc.2004.03.004 |
| [16] | Maia, L. A.; Montefusco, E.; Pellacci, B., Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system, Communications in Contemporary Mathematics, 10, 05, 651-669 (2008) · Zbl 1159.35428 · doi:10.1142/s0219199708002934 |
| [17] | Noris, B.; Ramos, M., Existence and bounds of positive solutions for a nonlinear Schrödinger system, Proceedings of the American Mathematical Society, 138, 5, 1681-1692 (2010) · Zbl 1189.35086 · doi:10.1090/s0002-9939-10-10231-7 |
| [18] | Noris, B.; Terracini, S.; Tavares, H.; Verzini, G., Uniform hölder bounds for nonlinear Schrödinger systems with strong competition, Communications on Pure and Applied Mathematics, 63, 3, 267-302 (2010) · Zbl 1189.35314 · doi:10.1002/cpa.20309 |
| [19] | Sato, Y.; Wang, Z.-Q., On the least energy sign-changing solutions for a nonlinear elliptic sytem, Discrete and Continuous Dynamical Systems, 35, 5, 2151-2164 (2014) · Zbl 1306.35027 · doi:10.3934/dcds.2015.35.2151 |
| [20] | Tavares, H.; Terracini, S., Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 29, 2, 279-300 (2012) · Zbl 1241.35046 · doi:10.1016/j.anihpc.2011.10.006 |
| [21] | Terracini, S.; Verzini, G., Multipulse phases in k-mixtures of Bose-Einstein condensates, Archive for Rational Mechanics and Analysis, 194, 3, 717-741 (2009) · Zbl 1181.35069 · doi:10.1007/s00205-008-0172-y |
| [22] | Wei, J.; Weth, T., Radial solutions and phase separation in a system of two coupled Schrödinger equations, Archive for Rational Mechanics and Analysis, 190, 1, 83-106 (2008) · Zbl 1161.35051 · doi:10.1007/s00205-008-0121-9 |
| [23] | Yue, X., Positive ground state solutions and multiple nontrivial solutions for coupled critical elliptic systems, Journal of Mathematical Analysis and Applications, 427, 1, 88-106 (2015) · Zbl 1333.35050 · doi:10.1016/j.jmaa.2015.01.073 |
| [24] | Chen, Z.; Zou, W., Positive least energy solutions and phase separation for coupled nonlinear Schrödinger equations with critical exponent, Archive for Rational Mechanics and Analysis, 205, 2, 515-551 (2012) · Zbl 1256.35132 · doi:10.1007/s00205-012-0513-8 |
| [25] | Chen, Z.; Zou, W., Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent: higher dimensional case, Calculus of Variations and Partial Differential Equations, 52, 1-2, 423-467 (2015) · Zbl 1312.35158 · doi:10.1007/s00526-014-0717-x |
| [26] | Kim, S., On vector solutions for coupled nonlinear Schrödinger equations with critical exponents, Communications on Pure & Applied Analysis, 12, 3, 1259-1277 (2013) · Zbl 1309.35137 · doi:10.3934/cpaa.2013.12.1259 |
| [27] | Chen, Z.; Lin, C.-S.; Zou, W., Sign-changing solutions and phase seperation for an elliptic system with critical exponent, Communications in Partial Differential Equations, 39, 10, 1827-1859 (2014) · Zbl 1308.35084 · doi:10.1080/03605302.2014.908391 |
| [28] | Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Communications on Pure and Applied Mathematics, 36, 4, 437-477 (1983) · Zbl 0541.35029 · doi:10.1002/cpa.3160360405 |
| [29] | Cerami, G.; Solimini, S.; Struwe, M., Some existence results for superlinear elliptic boundary value problems involving critical exponents, Journal of Functional Analysis, 69, 3, 289-306 (1986) · Zbl 0614.35035 · doi:10.1016/0022-1236(86)90094-7 |
| [30] | Ambrosetti, A.; Struwe, M., A note on the problem \(-\Deltau=\lambdau+u \left| u\right|^{2^\ast - 2}\), Manuscripta Mathematica, 54, 4, 373-379 (1986) · Zbl 0596.35043 · doi:10.1007/bf01168482 |
| [31] | Capozzi, A.; Fortunato, D.; Palmieri, G., An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Annales de l’Institut Henri Poincare (C) Non Linear Analysis, 2, 6, 463-470 (1985) · Zbl 0612.35053 · doi:10.1016/s0294-1449(16)30395-x |
| [32] | Miranda, C., Un’osservazione sul teorem di Brouwer, Bolletino della Unione Matematica Italiana Serie II, 3, 5-7 (1940) · JFM 66.0217.01 |
| [33] | Aubin, T., Problems isoperimetriques et espaces de Sobolev, Journal of Differential Geometry, 11, 4, 573-598 (1976) · Zbl 0371.46011 · doi:10.4310/jdg/1214433725 |
| [34] | Talenti, G., Best constants in Sobolev inequality, Annali di Matematica Pura ed Applicata, 110, 1, 352-372 (1976) · Zbl 0353.46018 · doi:10.1007/bf02418013 |
| [35] | Willem, M., Minimax Theorems (1996), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 0856.49001 |
| [36] | Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theory and applications, Journal of Functional Analysis, 14, 4, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7 |
| [37] | Nadezhda, R.; Tsvetomir, T.; Mikhail, K., Deformation lemma, ljusternik-Schnirellmann theory and mountain pass theorem on \(C^1\)-finsler manifolds, Serdica Mathematical Journal, 21, 239-266 (1995) · Zbl 0837.58009 |
| [38] | Solimini, S., On the solvability of some elliptic partial differential equations with the linear part at resonance, Journal of Mathematical Analysis and Applications, 117, 1, 138-152 (1986) · Zbl 0634.35030 · doi:10.1016/0022-247x(86)90253-2 |
| [39] | Yue, X.; Zou, W., Remarks on a Brezis-Nirenberg’s result, Journal of Mathematical Analysis and Applications, 425, 2, 900-910 (2015) · Zbl 1312.35104 · doi:10.1016/j.jmaa.2015.01.011 |
| [40] | Chen, Z.; Zou, W., On the Brezis-Nirenberg problem in a ball, Differential Integral Equations, 25, 5-6, 527-542 (2012) · Zbl 1265.35072 |
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