×

Infinitely many solutions for systems of \(n\) fourth order partial differential equations coupled with Navier boundary conditions. (English) Zbl 1304.49013

Summary: In this paper, the existence of infinitely many solutions for a class of systems of \(n\) fourth order partial differential equations coupled with Navier boundary conditions is established. The approach is fully based on B. Ricceri’s variational principle [J. Comput. Appl. Math. 113, No. 1–2, 401–410 (2000; Zbl 0946.49001)].

MSC:

49J27 Existence theories for problems in abstract spaces
49J45 Methods involving semicontinuity and convergence; relaxation
35J40 Boundary value problems for higher-order elliptic equations
35A15 Variational methods applied to PDEs
35J60 Nonlinear elliptic equations

Citations:

Zbl 0946.49001
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Afrouzi, G. A.; Heidarkhani, S.; O’Regan, D., Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem, Taiwanese J. Math., 15, 1, 201-210 (2011) · Zbl 1234.34016
[2] Bonanno, G.; D’Aguì, G., On the Neumann problem for elliptic equations involving the \(p\)-Laplacian, J. Math. Anal. Appl., 358, 223-228 (2009) · Zbl 1177.35069
[3] Bonanno, G.; Di Bella, B., A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343, 1166-1176 (2008) · Zbl 1145.34005
[4] Bonanno, G.; Di Bella, B., Infinitely many solutions for a fourth-order elastic beam equation, Nonlinear Differ. Equ. Appl. NoDEA, 18, 357-368 (2011) · Zbl 1222.34023
[5] Bonanno, G.; Di Bella, B., A fourth-order boundary value problem for a Sturm-Liouville type equation, Appl. Math. Comput., 217, 3635-3640 (2010) · Zbl 1210.34026
[6] Bonanno, G.; Molica Bisci, G., A remark on perturbed elliptic Neumann problems, Stud. Univ. Babeş-Bolyai Math., LV, 4 (2010) · Zbl 1265.35063
[7] Bonanno, G.; Molica Bisci, G., Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Bound. Value Probl., 2009, 1-20 (2009) · Zbl 1177.34038
[8] Bonanno, G.; Molica Bisci, G., Infinitely many solutions for a Dirichlet problem involving the \(p\)-Laplacian, Proc. R. Soc. Edinburgh, 140A, 737-752 (2010) · Zbl 1197.35125
[10] Bonanno, G.; Molica Bisci, G.; Raˇdulescu, V., Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz-Sobolev spaces, C.R. Acad. Sci. Paris Ser. I, 349, 263-268 (2011) · Zbl 1211.35110
[11] Bonanno, G.; Molica Bisci, G.; O’Regan, D., Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Model., 52, 152-160 (2010) · Zbl 1201.35102
[12] Bonanno, G.; Di Bella, B.; O’Regan, D., Non-trivial solutions for nonlinear fourth-order elastic beam equations, Comput. Math. Appl., 62, 1862-1869 (2011) · Zbl 1231.74259
[13] Candito, P., Infinitely many solutions to the Neumann problem for elliptic equations involving the \(p\)-Laplacian and with discontinuous nonlinearities, Proc. Edinburgh Math. Soc., 45, 397-409 (2002) · Zbl 1035.35040
[14] Candito, P.; Livrea, R., Infinitely many solutions for a nonlinear Navier boundary value problem involving the \(p\)-biharmonic, Stud. Univ. “Babeş-Bolyai Math., LV, 4 (2010) · Zbl 1249.35087
[15] Dai, G., Infinitely many solutions for a Neumann-type differential inclusion problem involving the \(p(x)\)-Laplacian, Nonlinear Anal., 70, 2297-2305 (2009) · Zbl 1170.35561
[16] Fan, X.; Ji, C., Existence of infinitely many solutions for a Neumann problem involving the \(p(x)\)-Laplacian, J. Math. Anal. Appl., 334, 248-260 (2007) · Zbl 1157.35040
[18] Graef, J. R.; Heidarkhani, S.; Kong, L., Multiple solutions for a class of \((p_1\),…,\(p_n)\)-biharmonic systems, Commun. Pure Appl. Anal. (CPAA), 12, 3, 1393-1406 (2013) · Zbl 1268.35049
[20] Heidarkhani, S.; Tian, Y.; Tang, C.-L., Existence of three solutions for a class of \((p_1\),…,\(p_n)\)-biharmonic systems with Navier boundary conditions, Ann. Polon. Math., 104, 261-277 (2012) · Zbl 1255.35103
[21] Kristály, A., Infinitely many solutions for a differential inclusion problem in \(R^N\), J. Differ. Equ., 220, 511-530 (2006) · Zbl 1194.35523
[22] Lazer, A. C.; McKenna, P. J., Large amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32, 537-578 (1990) · Zbl 0725.73057
[23] Li, C., The existence of infinitely many solutions of a class of nonlinear elliptic equations with a Neumann boundary conditions for both resonance and oscillation problems, Nonlinear Anal., 54, 431-443 (2003) · Zbl 1126.35320
[24] Li, C.; Li, S., Multiple solutions and sign-changing solutions of a class of nonlinear elliptic equations with Neumann boundary conditions, J. Math. Anal. Appl., 298, 14-32 (2004) · Zbl 1127.35014
[25] Li, L.; Tang, C.-L., Existence of three solutions for \((p,q)\)-biharmonic systems, Nonlinear Anal., 73, 796-805 (2010) · Zbl 1195.35137
[26] Li, C.; Tang, C.-L., Three solutions for a Navier boundary value problem involving the \(p\)-biharmonic, Nonlinear Anal., 72, 1339-1347 (2010) · Zbl 1180.35210
[27] Liu, H.; Su, N., Existence of three solutions for a \(p\)-biharmonic problem, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 15, 3, 445-452 (2008) · Zbl 1168.35342
[28] Marano, S.; Motreanu, D., Infinitely many critical points of non-differentiable functions and applications to the Neumann-type problem involving the \(p\)-Laplacian, J. Differ. Equ., 182, 108-120 (2002) · Zbl 1013.49001
[29] Omari, P.; Zanolin, F., An elliptic problem with arbitrarily small positive solutions, Nonlinear Differ. Equ. Electron. J. Differ. Equ. Conf., 05, 301-308 (2000) · Zbl 0959.35059
[30] Omari, P.; Zanolin, F., Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Commun. Partial Differ. Equ., 21, 5-6 (1996)
[31] Ricceri, B., A general variational principle and some of its applications, J. Comput. Appl. Math., 113, 401-410 (2000) · Zbl 0946.49001
[32] Ricceri, B., Infinitely many solutions of the Neumann problem for elliptic equations involving the \(p\)-Laplacian, Bull. London Math. Soc., 33, 3, 331-340 (2001) · Zbl 1035.35031
[33] Simon, J., Regularitè de la solution d’une equation non lineaire dans \(R^N\), (Bénilan, P.; Robert, J., Journes d’Analyse Non Linaire (Proc. Conf., Besanon, 1977). Journes d’Analyse Non Linaire (Proc. Conf., Besanon, 1977), Lecture Notes in Mathematics, vol. 665 (1978), Springer: Springer Berlin, Heidelberg, New York), 205-227
[34] Tehrani, H. T., Infinitely many solutions for indefinite semilinear elliptic equations without symmetry, Commun. Partial Differ. Equ., 21, 541-557 (1996) · Zbl 0855.35044
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.