The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. (English) Zbl 1214.35077
Let \(\Omega\subset {\mathbb R}^N\) be a bounded domain with smooth boundary and assume that \(h:\overline\Omega\times {\mathbb R}\rightarrow{\mathbb R}\) is a continuous function. Assume \(a\) and \(b\) are positive numbers and denote \(M(t):=at+b\). This paper is concerned with the qualitative analysis of solutions to the nonlinear stationary problem
\[ -M\left(\int_\Omega |\nabla u|^2dx\right)\Delta u=h(x,u)\qquad x\in\Omega\,, \]
subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\).
The main objective of the present paper is to establish the existence of multiple solutions to this class of Kirchhoff-type problems. This is done by means of variational methods which combine the qualitative analysis on Nehari manifolds with the fibering map method.
\[ -M\left(\int_\Omega |\nabla u|^2dx\right)\Delta u=h(x,u)\qquad x\in\Omega\,, \]
subject to the Dirichlet boundary condition \(u=0\) on \(\partial\Omega\).
The main objective of the present paper is to establish the existence of multiple solutions to this class of Kirchhoff-type problems. This is done by means of variational methods which combine the qualitative analysis on Nehari manifolds with the fibering map method.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
MSC:
| 35R09 | Integro-partial differential equations |
| 35J60 | Nonlinear elliptic equations |
| 35J20 | Variational methods for second-order elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
References:
| [1] | Ambrosetti, A.; Azorero, G. J.; Peral, I., Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137, 219-242 (1996) · Zbl 0852.35045 |
| [2] | Ambrosetti, A.; Brezis, H.; Cerami, G., Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122, 519-543 (1994) · Zbl 0805.35028 |
| [3] | Alves, C. O.; Corrêa, F. J.S. A.; Ma, T. F., Positive solutions for a Quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl., 49, 85-93 (2005) · Zbl 1130.35045 |
| [4] | Adimurthy; Pacella, F.; Yadava, L., On the number of positive solutions of some semilinear Dirichlet problems in a ball, Differential Integral Equations, 10, 1157-1170 (1997) · Zbl 0940.35069 |
| [5] | Bensedki, A.; Bouchekif, M., On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity, Math. Comput. Modelling, 49, 1089-1096 (2009) · Zbl 1165.35364 |
| [6] | Binding, P. A.; Drabek, P.; Huang, Y. X., On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations, 5, 1-11 (1997) · Zbl 0886.35060 |
| [7] | Brown, K. J.; Wu, T. F., A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations, 69, 1-9 (2007) · Zbl 1133.35337 |
| [8] | Brown, K. J.; Wu, T. F., A fibering map approach to a potential operator equation and its applications, Differential Integral Equations, 22, 1097-1114 (2009) · Zbl 1240.35569 |
| [9] | Brown, K. J.; Zhang, Y., The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193, 481-499 (2003) · Zbl 1074.35032 |
| [10] | Chipot, M.; Lovat, B., Some remarks on non local elliptic and parabolic problems, Nonlinear Anal., 30, 4619-4627 (1997) · Zbl 0894.35119 |
| [11] | Corrêa, F. J.S. A., On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal., 59, 1147-1155 (2004) · Zbl 1133.35043 |
| [12] | Crandall, M. G.; Rabinowitz, P. H., Bifurcation perturbation of simple eigenvalues and linearized stability, Arch. Ration. Mech. Anal., 52, 161-180 (1973) · Zbl 0275.47044 |
| [13] | Damascelli, L.; Grossi, M.; Pacella, F., Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle, Ann. Inst. H. Poincaré Anal. Non Linéaire, 16, 631-652 (1999) · Zbl 0935.35049 |
| [14] | D’Ancona, P.; Shibata, Y., On global solvability of non-linear viscoelastic equations in the analytic category, Math. Methods Appl. Sci., 17, 477-489 (1994) · Zbl 0803.35091 |
| [15] | D’Ancona, P.; Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108, 247-262 (1992) · Zbl 0785.35067 |
| [16] | Drábek, P.; Pohozaev, S. I., Positive solutions for the \(p\)-Laplacian: application of the fibering method, Proc. Roy. Soc. Edinburgh Sect. A, 127, 703-726 (1997) · Zbl 0880.35045 |
| [17] | Du, Y., Exact multiplicity and \(S\)-shaped bifurcation curve for some semilinear elliptic problems from combustion theory, SIAM J. Math. Anal., 32, 707-733 (2000) · Zbl 0974.35036 |
| [18] | Ekeland, I., On the variational principle, J. Math. Anal. Appl., 17, 324-353 (1974) · Zbl 0286.49015 |
| [19] | de Figueiredo, D. G.; Gossez, J. P.; Ubilla, P., Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199, 452-467 (2003) · Zbl 1034.35042 |
| [20] | Keller, H. B., Lectures on Numerical Methods in Bifurcation Problems (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0656.65063 |
| [21] | Korman, P., On uniqueness of positive solutions for a class of semilinear equations, Discrete Contin. Dyn. Syst., 8, 865-871 (2002) · Zbl 1090.35082 |
| [22] | Kuo, Y.; Lin, W.; Shieh, S.; Wang, W., A minimal energy tracking method for non-radially symmetric solutions of coupled nonlinear Schrodinger equations, J. Comput. Phys., 228, 7941-7956 (2009) · Zbl 1175.65117 |
| [23] | 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 |
| [24] | Nishihara, K., On a global solution of some quasilinear hyperbolic equation, Tokyo J. Math., 7, 437-459 (1984) · Zbl 0586.35059 |
| [25] | Ouyang, T.; Shi, J., Exact multiplicity of positive solutions for a class of semilinear problem II, J. Differential Equations, 158, 94-151 (1999) · Zbl 0947.35067 |
| [26] | Tang, M., Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 133, 705-717 (2003) · Zbl 1086.35053 |
| [27] | Wang, S. H., Rigorous analysis and estimates of \(S\)-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws, Proc. R. Soc. Lond. Ser. A, 454, 1031-1048 (1998) · Zbl 0917.34028 |
| [28] | Willem, M., Minimax Theorems (1996), Birkhäuser: Birkhäuser Boston · Zbl 0856.49001 |
| [29] | Wu, T. F., On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318, 253-270 (2006) · Zbl 1153.35036 |
| [30] | Wu, T. F., Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, Rocky Mountain J. Math., 39, 995-1012 (2009) · Zbl 1179.35129 |
| [31] | Zhang, Z.; Perera, K., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317, 456-463 (2006) · Zbl 1100.35008 |
| [32] | Zhao, Y.; Wang, Y.; Shi, J., Exact multiplicity of solutions and \(S\)-shaped bifurcation curve for a class of semilinear elliptic equations, J. Math. Anal. Appl., 331, 263-278 (2007) · Zbl 1121.34029 |
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.
