SIGN-CHANGING SOLUTIONS OF (e1, B)-LIMIT INCREASING OPERATOR EQUATION*

XIAN, XU

Glasgow mathematical journal, Tome 53 (2011) no. 3, pp. 535-554

Résumé

In this paper, by using the fixed point index method first we obtain some existence and multiplicity results for sign-changing solutions of an (e1, B)-limit increasing operator equation. The main results can be applied to many non-linear boundary value problems to obtain the existence and multiplicity results for sign-changing solutions. We also give a clear description of locations of these sign-changing solutions through strict lower and upper solutions. As an example, in the last section we obtain some existence and multiplicity results for sign-changing solutions of some Sturm–Liouville differential boundary value problems.

Détail

DOI : 10.1017/S0017089511000115

Mots-clés : 47H07, 47H10

Bibliographie

[1] 1.Bartsch, T. and Wang, Z. Q., Sign changing solutions of nonlinear Schrodinger equations, Topo. Meth. Nonlinear Anal. 13 (1999), 191–198. Google Scholar | DOI

[2] 2.Dancer, E. N. and Du, Y., Existence of changing sign solutions for some semilinear problems with jumping nonlinearities at zero, Proc. R. Soc. Edinb. 124 A (1994), 1165–1176. Google Scholar | DOI

[3] 3.Dancer, E. N. and Du, Y., On sign-changing solutions of certain semilinear elliptic problems, Appl. Anal. 56 (1995), 193–206. Google Scholar | DOI

[4] 4.Bartsch, T., Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), 117–152. Google Scholar | DOI

[5] 5.Bartsch, T. and Wang, Z. Q., On the existence of sign-changing solutions for semilinear Dirichlet problems, Topo. Meth. Nonlinear Anal. 7 (1996), 115–131. Google Scholar | DOI

[6] 6.Liu, Z., Localized critical points in Banach spaces and sign changing solutions of nonlinear p-Laplacian equations. Topological methods, variational methods (World Scientific Press, New Jersery, 2002). Google Scholar

[7] 7.Bartsch, T., Chang, K. C. and Wang, Z. Q., On the Morse indices of sign changing solutions of nonlinear elliptic problems, Math. Z. 233 (2000), 655–677. Google Scholar | DOI

[8] 8.Xian, X. and Jingxian, S., On sign-changing solution for some three-point boundary value problems, Nonlinear Anal. 59 (2004), 491–505. Google Scholar | DOI

[9] 9.Xian, X., Multiple sign-changing solutions for some m-point boundary value problems, Electron. J. Differ. Equ. 2004 (89) (2004), 1–14. Google Scholar

[10] 10.Xian, X. and O'Regan, D., Multiplicity of sign-changing solutions for some four-point boundary value problem, Nonlinear Anal. 69 (2008), 434–447. Google Scholar

[11] 11.Xian, X., Jingxian, S. and O'Regan, D., Nodal solutions for m-point boundary value problems using bifurcation methods, Nonlinear Anal. 68 (2008), 3034–3046. Google Scholar

[12] 12.Wenming, Z. and Schechter, M., Critical points theory and its applications (Springer Verlag, New York, NY, 2006). Google Scholar

[13] 13.Xian, X. and Jingxian, S., Solutions for an operator equation under the conditions of pairs of paralleled lower and upper solutions, Nonlinear Anal. 69 (2008), 2251–2266. Google Scholar | DOI

[14] 14.Amann, H., On the number of solutions of nonlinear equations in ordered Banach spaces, J. Funct. Anal. 11 (1972), 346–384. Google Scholar | DOI

[15] 15.Li, F.. Solutions of nonlinear operator equations and applications. PhD Thesis (Shandong University, 1996). Google Scholar

[16] 16.Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709. Google Scholar | DOI

[17] 17.Deimling, K., Nonlinear functional analysis (Springer Verlag, New York, NY, 1985). Google Scholar | DOI

[18] 18.Jingxian, S. and Xian, X., Three solution theorems for nonlinear operator equations and applications, J. Syst. Sci. Complex 18 (1) (2005), 119–125. Google Scholar

[19] 19.Dajun, G., Nonlinear functional analysis and applications (Beijing Sci. & Tec. Press, Beijing, China, 1994). Google Scholar

[20] 20.Dajun, G. and Lakshmikantham, V., Nonlinear problems in abstract cones (Academic Press, New York, NY, 1988). Google Scholar

[21] 21.Carla, S. and Motreanu, D., Constant-sign and sign-changing solutions for nonlinear eigenvalue problems, Nonlinear Anal. 68 (2008) 2668–2676. Google Scholar | DOI

[22] 22.Li, Y. and Liu, Z. L., Multiple and sign-changing solutions of an elliptic eigenvalue problem with constraint, Sci. China (Series A), 44 (1) (2001), 48–57. Google Scholar | DOI

[23] 23.Wang, Z.-Q., Sign-changing solutions for a class of nonlinear elliptic problems, in Nonlinear analysis (Chang, K.-C. and Long, Y., Editors), Nankai Series in Pure and Applied Math. 6 (2000), 370–383. Google Scholar

[24] 24.Zhang, Z. and Li, S., On sign-changing and multiple solutions of the p-Laplacian, J. Funct. Anal. 197 (2003), 447–468. Google Scholar | DOI

[25] 25.Zhang, Z. and Perera, K., Sign-changing solutions of Kirchhof-type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006), 456–463. Google Scholar | DOI

[26] 26.Shujie, L. and Zhi-Qiang, W., Ljusternik–Schnirelman theory in partially ordered Hilbert spaces, Trans. Amer. Math. Soc. 354 (8) (2002), 3207–3227. Google Scholar

[27] 27.Shujie, L. and Zhi-Qiang, W., Mountain pass theorem in order intervals and multiple solutions for semilinear elliptic Dirichlet problems, J. Anal. Math. 81 (2000), 373–396. Google Scholar

[28] 28.Wenming, Z., Sign-changing critical point theory (Springer-Verlag, New York, NY, 2008). Google Scholar

[29] 29.Rabinowitz, P. H., Some global results for nonlinear eigenvalues, J. Funct. Anal. 7 (1971), 487–513. Google Scholar | DOI

[30] 30.Rabinowitz, P. H., On bifurcation from infinity, J. Differ. Equ. 14 (1973), 462–475. Google Scholar | DOI