×

Energy estimates for seminodal solutions to an elliptic system with mixed couplings. (English) Zbl 1511.35144

Some semilinear elliptic systems with mixed couplings on the whole space \(\mathbb{R}^N\) are studied in this paper. Sufficient conditions are given in order to obtain fully nontrivial (i.e., with all nonzero components) solutions in the Sobolev space \(H^1(\mathbb{R}^N)\). Upper estimates to the energy of the system are also obtained, they are a useful auxiliary tool all along the paper. The so-called unimodal or semi-positive solutions are especially interesting. The method of proof uses symmetries combined with concentration-compactness arguments. Also, solutions with positive and non-radial signs changing components to an associated singularly perturbed system on the unit ball are exhibited.

MSC:

35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Ackermann, N.; Dancer, N., Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity, Differ. Integr. Equ., 29, 7-8, 757-774 (2016) · Zbl 1389.35071
[2] Bracho, J.; Clapp, M.; Marzantowicz, W., Symmetry breaking solutions of nonlinear elliptic systems, Topol. Methods Nonlinear Anal., 26, 1, 189-201 (2005) · Zbl 1152.35030
[3] Byeon, J.; Sato, Y.; Zhi-Qiang, W., Pattern formation via mixed attractive and repulsive interactions for nonlinear Schrödinger systems, J. Math. Pures Appl., 106, 3, 477-511 (2016) · Zbl 1345.35032
[4] Cerami, G.; Clapp, M., Sign changing solutions of semilinear elliptic problems in exterior domains, Calc. Var. Partial Differ. Equ., 30, 3, 353-367 (2007) · Zbl 1174.35026
[5] Chen, Z.; Chang-Shou, L.; Wenming, Z., Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrödinger system., Ann. Sc. Norm. Super Pisa Cl. Sci., 15, 859-897 (2016) · Zbl 1343.35101
[6] Cingolani, S.; Clapp, M.; Secchi, S., Multiple solutions to a magnetic nonlinear Choquard equation, Z. Angew. Math. Phys., 63, 2, 233-248 (2012) · Zbl 1247.35141
[7] Clapp, M.; Pistoia, A., Fully nontrivial solutions to elliptic systems with mixed couplings, Nonlinear Anal., 216 (2022) · Zbl 1481.35168
[8] Clapp, M., Soares, M.: Coupled and uncoupled sign-changing spikes of singularly perturbed elliptic systems. Communication in Contemporary Mathematics, to appear. arXiv:2108.00299 (2021) · Zbl 07729669
[9] Clapp, M.; Srikanth, PN, Entire nodal solutions of a semilinear elliptic equation and their effect on concentration phenomena, J. Math. Anal. Appl., 437, 1, 485-497 (2016) · Zbl 1334.35056
[10] Dovetta, S.; Pistoia, A., Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime, Math. Eng., 4, 21 (2022) · Zbl 1497.35173
[11] Esry, BD; Greene, CH; Burke, J.; James, P.; Bohn, JL, Hartree-Fock theory for double condensates, Phys. Rev. Lett., 78, 3594-3597 (1997)
[12] Lin, T-C; Wei, J., Spikes in two coupled nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22, 4, 403-439 (2005) · Zbl 1080.35143
[13] Lin, T-C; Wei, J., Ground state of \(N\) coupled nonlinear Schrödinger equations, Comm. Math. Phys., 255, 3, 629-653 (2005) · Zbl 1119.35087
[14] Liu, J.; Liu, X.; Wang, Z., Multiple mixed states of nodal solutions for nonlinear Schrödinger systems, Calc. Var. Partial Differ. Equ., 52, 3-4, 586 (2015) · Zbl 1311.35291
[15] Sato, Y.; Wang, Z-Q, On the multiple existence of semi-positive solutions for a nonlinear Schrödinger system., Ann. Inst. H. Poincaré C Anal. Non Linéaire, 30, 1-22 (2013) · Zbl 1457.35071
[16] Sato, Y.; Wang, Z-Q, On the least energy sign-changing solutions for a nonlinear elliptic system, Discrete Contin. Dyn. Syst., 35, 5, 2151-2164 (2015) · Zbl 1306.35027
[17] Sato, Y.; Wang, Z-Q, Least energy solutions for nonlinear Schrödinger systems with mixed attractive and repulsive couplings, Adv. Nonlinear Stud., 15, 1, 1-22 (2015) · Zbl 1316.35269
[18] Sato, Y.; Wang, Z-Q, Multiple positive solutions for Schrödinger systems with mixed couplings, Calc. Var. Partial Differ. Equ., 54, 2, 1373-1392 (2015) · Zbl 1326.35131
[19] Sirakov, B., Least energy solitary waves for a system of nonlinear Schrödinger equations in Rn, Comm. Math. Phys., 271, 1, 199-221 (2007) · Zbl 1147.35098
[20] Soave, N., On existence and phase separation of solitary waves for nonlinear Schrödinger systems modelling simultaneous cooperation and competition, Calc. Var. Partial Differ. Equ., 53, 3-4, 689-718 (2015) · Zbl 1323.35166
[21] Soave, N.; Tavares, H., New existence and symmetry results for least energy positive solutions of Schrödinger systems with mixed competition and cooperation terms, J. Differ. Equ., 261, 1, 505-537 (2016) · Zbl 1337.35049
[22] Tavares, H.; You, S., Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: The critical case, Calc. Var. Partial Differ. Equ., 59, 35 (2020) · Zbl 1459.35145
[23] Tavares, H.; You, S.; Zou, W., Least energy positive solutions of critical Schrödinger systems with mixed competition and cooperation terms: The higher dimensional case, J. Funct. Anal., 283, 50 (2022) · Zbl 1496.35198
[24] Tavares, H.; Terracini, S., Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems., Ann. Inst. H. Poincaré C Anal. Non Linéaire, 29, 279-300 (2012) · Zbl 1241.35046
[25] Wei, J.; Wu, Y., Ground states of nonlinear Schrödinger systems with mixed couplings, J. Math. Pures Appl., 141, 50-88 (2020) · Zbl 1448.35176
[26] Willem, M.: Minimax theorems. Progress in Nonlinear Differential Equations and their Applications 24. (Birkhäuser Boston, Boston, 1996) · Zbl 0856.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.