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Existence of solutions for a modified nonlinear Schrödinger system. (English) Zbl 1271.35065

Summary: We are concerned with the following modified nonlinear Schrödinger system: \(-\Delta u + u - (1/2)u\Delta(u^2) = (2\alpha/(\alpha + \beta))|u|^{\alpha - 2}|v|^\beta u\), \(x \in \Omega\), \(-\Delta v + v - (1/2)v\Delta(v^2) = (2\beta/(\alpha + \beta))|u|^\alpha|v|^{\beta - 2}v\), \(x \in \Omega\), \(u = 0\), \(v = 0\), \(x \in \partial \Omega\), where \(\alpha > 2\), \(\beta > 2\), \(\alpha + \beta < 2 \cdot 2^\ast\), \(2^\ast = 2N/(N - 2)\) is the critical Sobolev exponent, and \(\Omega \subset \mathbb R^N (N \geq 3)\) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B09 Positive solutions to PDEs
35B20 Perturbations in context of PDEs
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[1] S. Kurihara, “Large-amplitude quasi-solitons in superfluid films,” Journal of the Physical Society of Japan, vol. 50, no. 10, pp. 3262-3267, 1981.
[2] E. W. Laedke, K. H. Spatschek, and L. Stenflo, “Evolution theorem for a class of perturbed envelope soliton solutions,” Journal of Mathematical Physics, vol. 24, no. 12, pp. 2764-2769, 1983. · Zbl 0548.35101
[3] H. S. Brandi, C. Manus, G. Mainfray, T. Lehner, and G. Bonnaud, “Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. I: paraxial approximation,” Physics of Fluids B, vol. 5, no. 10, pp. 3539-3550, 1993.
[4] X. L. Chen and R. N. Sudan, “Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse in underdense plasma,” Physical Review Letters, vol. 70, no. 14, pp. 2082-2085, 1993.
[5] A. de Bouard, N. Hayashi, and J. C. Saut, “Global existence of small solutions to a relativistic nonlinear Schrödinger equation,” Communications in Mathematical Physics, vol. 189, no. 1, pp. 73-105, 1997. · Zbl 0948.81025
[6] B. Ritchie, “Relativistic self-focusing and channel formation in laser-plasma interactions,” Physical Review E, vol. 50, no. 2, pp. R687-R689, 1994.
[7] M. Poppenberg, K. Schmitt, and Z. Q. Wang, “On the existence of soliton solutions to quasilinear Schrödinger equations,” Calculus of Variations and Partial Differential Equations, vol. 14, no. 3, pp. 329-344, 2002. · Zbl 1052.35060
[8] J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, “Soliton solutions for quasilinear Schrödinger equations II,” Journal of Differential Equations, vol. 187, no. 2, pp. 473-493, 2003. · Zbl 1229.35268
[9] M. Colin and L. Jeanjean, “Solutions for a quasilinear Schrödinger equation: a dual approach,” Nonlinear Analysis A, vol. 56, no. 2, pp. 213-226, 2004. · Zbl 1035.35038
[10] H. Berestycki and P. L. Lions, “Nonlinear scalar field equations. I: existence of a ground state,” Archive for Rational Mechanics and Analysis, vol. 82, no. 4, pp. 313-345, 1983. · Zbl 0533.35029
[11] J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, “Solutions for quasilinear Schrödinger equations via the Nehari method,” Communications in Partial Differential Equations, vol. 29, no. 5-6, pp. 879-901, 2004. · Zbl 1140.35399
[12] Y. Guo and Z. Tang, “Ground state solutions for quasilinear Schrödinger systems,” Journal of Mathematical Analysis and Applications, vol. 389, no. 1, pp. 322-339, 2012. · Zbl 1236.35036
[13] P. L. Lions, “The concentration-compactness principle in the calculus of variations. The locally compact case I,” Annales de l’Institut Henri Poincaré, vol. 1, no. 2, pp. 109-145, 1984. · Zbl 0541.49009
[14] Y. Guo and Z. Tang, “Ground state solutions for the quasilinear Schrödinger equation,” Nonlinear Analysis A, vol. 75, no. 6, pp. 3235-3248, 2012. · Zbl 1234.35246
[15] T. Bartsch and Z. Q. Wang, “Multiple positive solutions for a nonlinear Schrödinger equation,” Zeitschrift für Angewandte Mathematik und Physik, vol. 51, no. 3, pp. 366-384, 2000. · Zbl 0972.35145
[16] A. Ambrosetti, M. Badiale, and S. Cingolani, “Semiclassical states of nonlinear Schrödinger equations,” Archive for Rational Mechanics and Analysis, vol. 140, no. 3, pp. 285-300, 1997. · Zbl 0896.35042
[17] A. Ambrosetti, A. Malchiodi, and S. Secchi, “Multiplicity results for some nonlinear Schrödinger equations with potentials,” Archive for Rational Mechanics and Analysis, vol. 159, no. 3, pp. 253-271, 2001. · Zbl 1040.35107
[18] J. Byeon and Z. Q. Wang, “Standing waves with a critical frequency for nonlinear Schrödinger equations II,” Calculus of Variations and Partial Differential Equations, vol. 18, no. 2, pp. 207-219, 2003. · Zbl 1073.35199
[19] S. Cingolani and M. Lazzo, “Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions,” Journal of Differential Equations, vol. 160, no. 1, pp. 118-138, 2000. · Zbl 0952.35043
[20] S. Cingolani and M. Nolasco, “Multi-peak periodic semiclassical states for a class of nonlinear Schrödinger equations,” Proceedings of the Royal Society of Edinburgh A, vol. 128, no. 6, pp. 1249-1260, 1998. · Zbl 0922.35158
[21] M. Del Pino and P. L. Felmer, “Multi-peak bound states for nonlinear Schrödinger equations,” Annales de l’Institut Henri Poincaré, vol. 15, no. 2, pp. 127-149, 1998. · Zbl 0901.35023
[22] M. del Pino and P. L. Felmer, “Semi-classical states for nonlinear Schrödinger equations,” Journal of Functional Analysis, vol. 149, no. 1, pp. 245-265, 1997. · Zbl 0887.35058
[23] A. Floer and A. Weinstein, “Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential,” Journal of Functional Analysis, vol. 69, no. 3, pp. 397-408, 1986. · Zbl 0613.35076
[24] Y. G. Oh, “On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential,” Communications in Mathematical Physics, vol. 131, no. 2, pp. 223-253, 1990. · Zbl 0753.35097
[25] Y. G. Oh, “Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class (V)a,” Communications in Partial Differential Equations, vol. 13, no. 12, pp. 1499-1519, 1988. · Zbl 0702.35228
[26] X. Q. Liu, J. Q. Liu, and Z. Q. Wang, “Quasilinear elliptic equations via perturbation method,” Proceedings of the American Mathematical Society, vol. 141, no. 1, pp. 253-263, 2013. · Zbl 1267.35096
[27] J. Liu and Z. Q. Wang, “Soliton solutions for quasilinear Schrödinger equations I,” Proceedings of the American Mathematical Society, vol. 131, no. 2, pp. 441-448, 2003. · Zbl 1229.35269
[28] A. Canino and M. Degiovanni, “Nonsmooth critical point theory and quasilinear elliptic equations,” in Topological Methods in Differential Equations and Inclusions, P. Q. Montreal, Ed., vol. 472 of Nato Advanced Study Institute Series C: Mathematical and Physical Sciences, pp. 1-50, Kluwer Academic, 1995. · Zbl 0851.35038
[29] J. Q. Liu and Z. Q. Wang, “Bifurcations for quasilinear elliptic equations II,” Communications in Contemporary Mathematics, vol. 10, no. 5, pp. 721-743, 2008. · Zbl 1165.35037
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