In this paper, we investigate the nonlinear Schrödinger-Kirchhoff equations on the whole space. By using the Morse index of the reduced Schrödinger operator, we show the existence and multiplicity of solutions for this problem with asymptotically linear nonlinearity via variational methods.
Citation: Yuan Shan, Baoqing Liu. Existence and multiplicity of solutions for generalized asymptotically linear Schrödinger-Kirchhoff equations[J]. AIMS Mathematics, 2021, 6(6): 6160-6170. doi: 10.3934/math.2021361
In this paper, we investigate the nonlinear Schrödinger-Kirchhoff equations on the whole space. By using the Morse index of the reduced Schrödinger operator, we show the existence and multiplicity of solutions for this problem with asymptotically linear nonlinearity via variational methods.
| [1] |
A. Ambrosetti, P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381. doi: 10.1016/0022-1236(73)90051-7
|
| [2] |
T. Bartsch, A. Pankov, Z. Q. Wang, Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3 (2001), 549-569. doi: 10.1142/S0219199701000494
|
| [3] |
Y. Ding, L. Jeanjean, Homoclinc orbits for a nonperiodic Hamiltonian system, J. Differ. Equations, 237 (2007), 473-490. doi: 10.1016/j.jde.2007.03.005
|
| [4] | I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Berlin: Springer-Verlag, 1990. |
| [5] |
X. He, W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70 (2009), 1407-1414. doi: 10.1016/j.na.2008.02.021
|
| [6] |
X. He, W. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 26 (2010), 387-394. doi: 10.1007/s10255-010-0005-2
|
| [7] |
X. He, W. Zou, Ground states for nonlinear Kirchhoff equations with critical growth, Ann. Mat. Pura Appl., 193 (2014), 473-500. doi: 10.1007/s10231-012-0286-6
|
| [8] |
J. Jin, X. Wu, Infinitely many radial solutions for Kirchhoff-type problems in $ {\mathbb R}^N$, J. Math. Anal. Appl., 369 (2010), 564-574. doi: 10.1016/j.jmaa.2010.03.059
|
| [9] | G. Kirchhoff, Mechanik, Teubner, Leipzig: Dbuck Und Verlag Von B. G. Teubner, 1897. |
| [10] |
L. Li, J. J. Sun, Existence and multiplicity of solutions for the Kirchhoff equations with asymptotically linear nonlinearities, Nonlinear Anal. Real World Appl., 26 (2015), 391-399. doi: 10.1016/j.nonrwa.2015.07.002
|
| [11] |
Q. Li, X. Wu, A new result on high energy solutions for Schrödinger-Kirchhoff type equations in $ {\mathbb R}^N$, Appl. Math. Lett., 30 (2014), 24-27. doi: 10.1016/j.aml.2013.12.002
|
| [12] |
Y. Li, F. Li, J. Shi, Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differ. Equations, 253 (2012), 2285-2294. doi: 10.1016/j.jde.2012.05.017
|
| [13] |
J. L. Lions, On some questions in boundary value problems of mathematical physics, North-Holland Math. Stud., 30 (1978), 284-346. doi: 10.1016/S0304-0208(08)70870-3
|
| [14] |
Z. Liu, J. Su, T. Weth, Compactness results for Schrödinger equations with asymptotically linear terms, J. Differ. Equations, 231 (2006), 501-512. doi: 10.1016/j.jde.2006.05.007
|
| [15] |
W. Liu, X. He, Multiplicity of high energy solutions for superlinear Kirchhoff equations, J. Appl. Math. Comput., 39 (2012), 473-487. doi: 10.1007/s12190-012-0536-1
|
| [16] | Y. Long, Index Theory for Symplectic Paths with Applications, Basel: Birkhäuser, 2002. |
| [17] |
T. F. Ma, J. E. Mu${{\rm{\bar n}}}$oz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett., 16 (2003), 243-248. doi: 10.1016/S0893-9659(03)80038-1
|
| [18] |
A. Mao, Z. Zhang, Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition, Nonlinear Anal., 70 (2009), 1275-1287. doi: 10.1016/j.na.2008.02.011
|
| [19] |
K. Perera, Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differ. Equations, 221 (2006), 246-255. doi: 10.1016/j.jde.2005.03.006
|
| [20] | M. Reed, B. Simon, Methods of Modern Mathematical Physics, New York: Academic Press, 1978. |
| [21] |
Y. Shan, Morse index and multiple solutions for the asymptotically linear Schrödinger type equation, Nonliear Anal., 89 (2013), 170-178. doi: 10.1016/j.na.2013.05.014
|
| [22] |
W. Shuai, Sign-changing solutions for a class of Kirchhoff-type problem in bounded domains, J. Differ. Equations, 259 (2015), 1256-1274. doi: 10.1016/j.jde.2015.02.040
|
| [23] |
B. Simon, Schrödinger semigroups, Bull. Am. Math. Soc., 7 (1982), 447-526. doi: 10.1090/S0273-0979-1982-15041-8
|
| [24] | M. Struwe, Variational Methods: Applications to Nonlinear Partifal Differential Equations and Hamiltonian Systems, Berlin: Springer, 1990. |
| [25] |
X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in $ {\mathbb R}^N$, Nonlinear Anal. Real World Appl., 12 (2011), 1278-1287. doi: 10.1016/j.nonrwa.2010.09.023
|
| [26] |
Y. Wu, S. Liu, Existence and multiplicity of solutions for asymptotically linear Schrödinger-Kirchhoff equations, Nonlinear Anal. Real World Appl., 26 (2015), 191-198. doi: 10.1016/j.nonrwa.2015.05.010
|
| [27] |
Z. Yücedaǧ, Solutions of nonlinear problems involving $p(x)$ Laplacian operator, Adv. Nonlinear Anal., 4 (2015), 285-293. doi: 10.1515/anona-2015-0044
|
| [28] |
F. Zhou, K. Wu, X. Wu, High energy solutions of systems of Kirchhoff-type equations on $ {\mathbb R}^N$, Comput. Math. Appl., 66 (2013), 1299-1305. doi: 10.1016/j.camwa.2013.07.028
|
Yuan Shan, Baoqing Liu. Existence and multiplicity of solutions for generalized asymptotically linear Schrödinger-Kirchhoff equations[J]. AIMS Mathematics, 2021, 6(6): 6160-6170. doi: 10.3934/math.2021361
DownLoad: