The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions

Journal of Differential Equations

Volumn 250, Issue 4, 2011, Pages 1876-1908

  Chen, Ching yu ^a   Kuo, Yueh cheng ^a   Wu, Tsung fang ^a  

^a NATIONAL UNIVERSITY OF KAOHSIUNG   (Taiwan)

Author keywords

Fibering map; Kirchhoff type equation; Nehari manifold; Sign changing weight

Indexed keywords

EID: 78650993948     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2010.11.017     Document Type: Article

References (32)

1. Ambrosetti A., Azorero G.J., Peral I. Multiplicity results for some nonlinear elliptic equations. J. Funct. Anal. 1996, 137:219-242.
2. Ambrosetti A., Brezis H., Cerami G. Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 1994, 122:519-543.
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. 2005, 49:85-93.
4. Adimurthy, Pacella F., Yadava L. On the number of positive solutions of some semilinear Dirichlet problems in a ball. Differential Integral Equations 1997, 10:1157-1170.
5. Bensedki A., Bouchekif M. On an elliptic equation of Kirchhoff-type with a potential asymptotically linear at infinity. Math. Comput. Modelling 2009, 49:1089-1096.
6. Binding P.A., Drabek P., Huang Y.X. On Neumann boundary value problems for some quasilinear elliptic equations. Electron. J. Differential Equations 1997, 5:1-11.
7. Brown K.J., Wu T.F. A fibering map approach to a semilinear elliptic boundary value problem. Electron. J. Differential Equations 2007, 69:1-9.
8. Brown K.J., Wu T.F. A fibering map approach to a potential operator equation and its applications. Differential Integral Equations 2009, 22:1097-1114.
9. Brown K.J., Zhang Y. The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function. J. Differential Equations 2003, 193:481-499.
10. Chipot M., Lovat B. Some remarks on non local elliptic and parabolic problems. Nonlinear Anal. 1997, 30:4619-4627.
11. Corrêa F.J.S.A. On positive solutions of nonlocal and nonvariational elliptic problems. Nonlinear Anal. 2004, 59:1147-1155.
12. Crandall M.G., Rabinowitz P.H. Bifurcation perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 1973, 52:161-180.
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 1999, 16:631-652.
14. D'Ancona P., Shibata Y. On global solvability of non-linear viscoelastic equations in the analytic category. Math. Methods Appl. Sci. 1994, 17:477-489.
15. D'Ancona P., Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 1992, 108:247-262.
16. Drábek P., Pohozaev S.I. Positive solutions for the p-Laplacian: application of the fibering method. Proc. Roy. Soc. Edinburgh Sect. A 1997, 127:703-726.
17. Du Y. Exact multiplicity and S-shaped bifurcation curve for some semilinear elliptic problems from combustion theory. SIAM J. Math. Anal. 2000, 32:707-733.
18. Ekeland I. On the variational principle. J. Math. Anal. Appl. 1974, 17:324-353.
19. de Figueiredo D.G., Gossez J.P., Ubilla P. Local superlinearity and sublinearity for indefinite semilinear elliptic problems. J. Funct. Anal. 2003, 199:452-467.
20. Keller H.B. Lectures on Numerical Methods in Bifurcation Problems 1987, Springer-Verlag, Berlin.
21. Korman P. On uniqueness of positive solutions for a class of semilinear equations. Discrete Contin. Dyn. Syst. 2002, 8:865-871.
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. 2009, 228:7941-7956.
23. Ma T.F., Muñoz Rivera J.E. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett. 2003, 16(2):243-248.
24. Nishihara K. On a global solution of some quasilinear hyperbolic equation. Tokyo J. Math. 1984, 7:437-459.
25. Ouyang T., Shi J. Exact multiplicity of positive solutions for a class of semilinear problem II. J. Differential Equations 1999, 158:94-151.
26. Tang M. Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities. Proc. Roy. Soc. Edinburgh Sect. A 2003, 133:705-717.
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 1998, 454:1031-1048.
28. Willem M. Minimax Theorems 1996, Birkhäuser, Boston.
29. Wu T.F. On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J. Math. Anal. Appl. 2006, 318:253-270.
30. Wu T.F. Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mountain J. Math. 2009, 39:995-1012.
31. Zhang Z., Perera K. Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 2006, 317:456-463.
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. 2007, 331:263-278.