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Existence and multiplicity of solutions for asymptotically linear Schrödinger-Kirchhoff equations. (English) Zbl 1334.35060
Summary: The purpose of this work is to study a Schrödinger-Kirchhoff equation in \(\mathbb R^3\) with the nonlinearity asymptotically linear and the potential indefinite in sign. By variational methods, we obtain the existence of multiple nontrivial solutions for this problem.

MSC:
35J61 Semilinear elliptic equations
35R09 Integral partial differential equations
35J35 Variational methods for higher-order elliptic equations
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[1] Kirchhoff, G., Mechanik, (1883), Teubner Leipzig · JFM 08.0542.01
[2] Lions, J. L., On some questions in boundary value problems of mathematical physics, (Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proceedings of International Symposium, Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977, North-Holland Math. Stud., vol. 30, (1978), North-Holland Amsterdam, New York), 284-346
[3] Arosio, A.; Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. Amer. Math. Soc., 348, 1, 305-330, (1996) · Zbl 0858.35083
[4] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Soriano, J. A., Global existence and uniform decay rates for the Kirchhoff-carrier equation with nonlinear dissipation, Adv. Differential Equations, 6, 6, 701-730, (2001) · Zbl 1007.35049
[5] D’Ancona, P.; Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108, 2, 247-262, (1992) · Zbl 0785.35067
[6] Sun, J.; Liu, S., Nontrivial solutions of Kirchhoff type problems, Appl. Math. Lett., 25, 3, 500-504, (2012) · Zbl 1251.35027
[7] Ma, T. F.; Muñoz Rivera, J. E., Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16, 2, 243-248, (2003) · Zbl 1135.35330
[8] Perera, K.; Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221, 1, 246-255, (2006) · Zbl 1357.35131
[9] Zhang, Z.; Perera, K., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317, 2, 456-463, (2006) · Zbl 1100.35008
[10] Mao, A.; Zhang, Z., Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70, 3, 1275-1287, (2009) · Zbl 1160.35421
[11] He, X.; Zou, W., Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 26, 3, 387-394, (2010) · Zbl 1196.35077
[12] He, X.; Zou, W., Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70, 3, 1407-1414, (2009) · Zbl 1157.35382
[13] Yang, Y.; Zhang, J., Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23, 4, 377-380, (2010) · Zbl 1188.35084
[14] Chen, S.; Liu, S., Standing waves for 4-superlinear Schrödinger-Kirchhoff equations, Math. Methods Appl. Sci., 1-9, (2014)
[15] He, X.; Zou, W., Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl. (4), 193, 2, 473-500, (2014) · Zbl 1300.35016
[16] Liu, W.; He, X., Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39, 1-2, 473-487, (2012) · Zbl 1295.35226
[17] Li, Y.; Li, F.; Shi, J., Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253, 7, 2285-2294, (2012) · Zbl 1259.35078
[18] Li, Q.; Wu, X., A new result on high energy solutions for Schrödinger-Kirchhoff type equations in \(\mathbb{R}^N\), Appl. Math. Lett., 30, 24-27, (2014) · Zbl 1317.35035
[19] Wu, X., Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in \(R^N\), Nonlinear Anal. RWA, 12, 2, 1278-1287, (2011) · Zbl 1208.35034
[20] Wu, X., High energy solutions of systems of Kirchhoff-type equations in \(R^N\), J. Math. Phys., 53, 6, 063508, (2012), 18. http://dx.doi.org/10.1063/1.4729543 · Zbl 1277.35336
[21] Jin, J.; Wu, X., Infinitely many radial solutions for Kirchhoff-type problems in \(\mathbb{R}^N\), J. Math. Anal. Appl., 369, 2, 564-574, (2010) · Zbl 1196.35221
[22] Zhou, F.; Wu, K.; Wu, X., High energy solutions of systems of Kirchhoff-type equations on \(\mathbb{R}^N\), Comput. Math. Appl., 66, 7, 1299-1305, (2013) · Zbl 1355.35066
[23] Reed, M.; Simon, B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, (1975), Academic Press New York, London · Zbl 0308.47002
[24] Reed, M.; Simon, B., Methods of modern mathematical physics. IV. analysis of operators, (1978), Academic Press New York, London · Zbl 0401.47001
[25] Willem, M., (Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, (1996), Birkhäuser Boston Inc. Boston, MA)
[26] Liu, J., The Morse index of a saddle point, J. Systems Sci. Math. Sci., 2, 1, 32-39, (1989) · Zbl 0732.58011
[27] Liu, J.; Su, J., Remarks on multiple nontrivial solutions for quasi-linear resonant problems, J. Math. Anal. Appl., 258, 1, 209-222, (2001) · Zbl 1050.35025
[28] Clark, D. C., A variant of the Lusternik-schnirelman theory, Indiana Univ. Math. J., 22, 65-74, (1972-1973) · Zbl 0228.58006
[29] Rabinowitz, P. H., (Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, (1986), Published for the Conference Board of the Mathematical Sciences Washington, DC) · Zbl 0609.58002
[30] Chen, S.; Wang, C., Existence of multiple nontrivial solutions for a Schrödinger-Poisson system, J. Math. Anal. Appl., 411, 2, 787-793, (2014) · Zbl 1331.35125
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