×

Infinitely many solutions for a class of Dirichlet quasilinear elliptic systems. (English) Zbl 1253.34026

From the introduction: The aim of this paper is to investigate the existence of infinitely many classical solutions for the following Dirichlet quasilinear elliptic system \[ -(p_i- 1)|u_i'(x)|^{p_i-2} u_i''(x)=\lambda F_{u_i}(x, u_1,\dots, u_n) h_i(x, u_i'),\quad x\in (a,b), \]
\[ u_i(a)= u_i(b)= 0,\quad\text{for }1\leq i\leq n, \] where \(p_i> 1\) for \(1\leq i\leq n\), \(\lambda\) is a positive parameter, \(a,b\in\mathbb{R}\) with \(a< b\), \(h_i:[a,b]\times \mathbb{R}\to[0,+\infty)\) is a bounded and continuous function with \(m_i:= \text{inf}_{(x,t)\in [a,b]\times \mathbb{R}} h_i(x,t)> 0\) for \(1\leq i\leq n\), \(F: [a,b]\times\mathbb{R}^n\to \mathbb{R}\) is a function such that the mapping \((t_1,t_2,\dots, t_n)\to F(x, t_1,t_2,\dots, t_n)\) is in \(C^1\) in \(\mathbb{R}^n\) for all \(x\in [a,b]\), \(F_{u_i}\) is continuous in \([a,b]\times \mathbb{R}^n\) for \(1\leq i\leq n\), where \(F_{u_i}\) denotes the partial derivative of \(F\) with respect to \(u_i\), and \[ \sup_{|(t_1,\dots, t_n)|\leq M}|F_{u_i}(x, t_1,\dots, t_n)|\in L^1([a,b]) \] for all \(M> 0\) and all \(1\leq i\leq n\).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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
[2] Bonanno, G.; D’Aguì, G., A Neumann boundary value problem for the Sturm-Liouville equation, Appl. Math. Comput., 208, 318-327 (2009) · Zbl 1176.34020
[3] Bonanno, G.; Di Bella, B., Infinitely many solutions for a fourth-order elastic beam equation, NoDEA Nonlinear Differential Equations Appl., 18, 357-368 (2011) · Zbl 1222.34023
[4] Bonanno, G.; Molica Bisci, G., Infinitely many solutions for a Dirichlet problem involving the \(p\)-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 140, 737-752 (2010) · Zbl 1197.35125
[5] Bonanno, G.; Molica Bisci, G.; O’Regan, D., Infinitely many weak solutions for a class of quasilinear elliptic systems, Math. Comput. Modelling, 52, 152-160 (2010) · Zbl 1201.35102
[6] Bonanno, G.; Molica Bisci, G.; Rădulescu, 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
[7] Bonanno, G.; Molica Bisci, G.; Rădulescu, V., Infinitely many solutions for a class of nonlinear elliptic problems on fractals, C. R. Acad. Sci. Paris, Ser. I, 350, 387-391 (2012)
[8] Bonanno, G.; Tornatore, E., Infinitely many solutions for a mixed boundary value problem, Ann. Polon. Math., 99, 285-293 (2010) · Zbl 1208.34021
[9] Candito, P.; Livrea, R., Infinitely many solutions for a nonlinear Navier boundary value problem involving the \(p\)-biharmonic, Stud. Univ. Babes-Bolyai Math., 55, 41-51 (2010) · Zbl 1249.35087
[10] Chung, N. T., Existence of infinitely many solutions for degenerate and singular elliptic systems with indefinite concave nonlinearities, Electron. J. Differential Equations, 30, 1-12 (2011) · Zbl 1220.35036
[11] Ricceri, B., Infinitely many solutions of the Neumann problem for elliptic equations involving the \(p\)-Laplacian, Bull. Lond. Math. Soc., 33, 331-340 (2001) · Zbl 1035.35031
[12] Graef, J. R.; Heidarkhani, S.; Kong, L., A critical points approach for the existence of multiple solutions of a Dirichlet quasilinear system, J. Math. Anal. Appl., 388, 1268-1278 (2012) · Zbl 1244.34024
[13] Ghergu, M.; Rădulescu, V., (Singular Elliptic Problems. Bifurcation and Asymptotic Analysis. Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, vol. 37 (2008), Oxford University Press) · Zbl 1159.35030
[14] Kristály, A.; Rădulescu, V.; Varga, C., (Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems. Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, vol. 136 (2010), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1206.49002
[15] Ricceri, B., Nonlinear eigenvalue problems, (Gao, D. Y.; Motreanu, D., Handbook of Nonconvex Analysis and Applications (2010), International Press), 543-595 · Zbl 1222.35137
[16] E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Berlin, Heidelberg, New York, 1990.; E. Zeidler, Nonlinear Functional Analysis and its Applications, vol. II/B, Berlin, Heidelberg, New York, 1990. · Zbl 0684.47029
[17] Ricceri, B., A general variational principle and some of its applications, J. Comput. Appl. Math., 113, 401-410 (2000) · Zbl 0946.49001
[18] Talenti, G., Some inequalities of Sobolev type on two-dimensional spheres, (Walter, W., General Inequalities, Vol. 5. General Inequalities, Vol. 5, Internat. Ser. Numer. Math., vol. 80 (1987), Birkhäuser: Birkhäuser Basel), 401-408
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.