Advanced Search
Article Contents
Article Contents

Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity

Abstract Related Papers Cited by
  • In this paper, we establish a unilateral global bifurcation result from interval for a class of $p$-Laplacian problems. By applying above result, we study the spectrum of a class of half-quasilinear problems. Moreover, we also investigate the existence of nodal solutions for a class of half-quasilinear eigenvalue problems.
    Mathematics Subject Classification: Primary: 34C10; Secondary: 34C23.


    \begin{equation} \\ \end{equation}
  • [1]

    A. Anane, O. Chakrone and M. Monssa, Spectrum of one dimensional $p$-Laplacian with indefinite weight, Electron. J. Qual. Theory Differ. Equ., 17 (2002), 11 pp.


    H. Berestycki, On some nonlinear Sturm-Liouville problems, J. Differential Equations, 26 (1977), 375-390.doi: 10.1016/0022-0396(77)90086-9.


    H. Brezis, Opérateurs Maximaux Monotone et Semigroup de Contractions dans les Espase de Hilbert, Math. Studies, 5, North-Holland, Amsterdam, 1973.


    G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468.doi: 10.1016/j.jde.2011.09.026.


    E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems, Indiana U. Math J., 23 (1974), 1069-1076.


    E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. Lond. Math. Soc., 34 (2002), 533-538.doi: 10.1112/S002460930200108X.


    M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian, J. Differential Equations, 92 (1991), 226-251.doi: 10.1016/0022-0396(91)90048-E.


    P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbbR^N$, Trans. Amer. Math. Soc., 349 (1997), 171-188.doi: 10.1090/S0002-9947-97-01788-1.


    J. Fleckinger and W. Reichel, Global solution branches for $p$-Laplacian boundary value problems, Nonlinear Anal., 62 (2005), 53-70.doi: 10.1016/j.na.2003.11.015.


    J. Fleckinger, R. Manásevich and Thélin, Global bifurcation from the first eigenvalue for a system of $p$-Laplacians, Math. Nachr., 182 (1996), 217-242.doi: 10.1002/mana.19961820110.


    J. García-Azorero and I. Peral, Existence and non-uniqueness for the $p$-Laplacian: Nonlinear eigenvalues, Comm. Part. Diff. Equations, 12 (1987), 1389-1430.doi: 10.1080/03605308708820534.


    J. García-Melián and J. Sabina de Lis, A local bifurcation theorem for degenerate elliptic equations with radial symmetry, J. Differential Equations, 179 (2002), 27-43.doi: 10.1006/jdeq.2001.4031.


    P. Girg and P. Takáč, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems, Ann. Henri Poincar'e, 9 (2008), 275-327.doi: 10.1007/s00023-008-0356-x.


    J. K. Hale, Bifurcation from simple eigenvalues for several parameter families, Nonlinear Anal., 2 (1978), 491-497.doi: 10.1016/0362-546X(78)90056-1.


    P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function, Comm. Partial Differential Equations, 5 (1980), 999-1030.doi: 10.1080/03605308008820162.


    B. Im, E. Lee and Y. H. Lee, A global bifurcation phenomena for second order singular boundary value problems, J. Math. Anal. Appl., 308 (2005), 61-78.doi: 10.1016/j.jmaa.2004.10.054.


    M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1965.


    T. Kusano, T. Jaros and N. Yoshida, A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonlinear Anal., 40 (2000), 381-395.doi: 10.1016/S0362-546X(00)85023-3.


    Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differential Equations, 229 (2006), 229-256.doi: 10.1016/j.jde.2006.03.021.


    J. López-Gómez, Multiparameter local bifurcation based on the linear part, J. Math. Anal. Appns., 138 (1989), 358-370.doi: 10.1016/0022-247X(89)90296-5.


    J. López-Gómez, Positive periodic solutions of Lotka-Volterra Systems, Diff. Int. Eqns., 5 (1992), 55-72.


    J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 2001.doi: 10.1201/9781420035506.


    J. López-Gómez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory, Nonlinear Anal., 58 (2004), 749-777.doi: 10.1016/j.na.2004.04.011.


    J. López-Gómez and C. Mora-Corral, Counting zeroes of $C^1$-Fredholm maps of index 1, Bull. Lond. Math. Soc., 37 (2005), 778-792.doi: 10.1112/S0024609305004716.


    J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators, Advances in Operator Theory and Applications, Vol. 177, Birkhaüser, Basel, 2007.


    R. Ma and B. Thompson, Nodal solutions for nonlinear eigenvalue problems, Nonlinear Anal., 59 (2004), 707-718.doi: 10.1016/j.na.2004.07.030.


    R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p$-Laplacian-like operators, J. Differential Equations, 145 (1998), 367-393.doi: 10.1006/jdeq.1998.3425.


    P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations, Commun. Pure Appl. Math., 23 (1970), 939-961.doi: 10.1002/cpa.3160230606.


    P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.doi: 10.1016/0022-1236(71)90030-9.


    P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475.doi: 10.1016/0022-0396(73)90061-2.


    P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math., 3 (1973), 161-202.doi: 10.1216/RMJ-1973-3-2-161.


    B. P. Rynne, Bifurcation from zero or infinity in Sturm-Liouville problems which are not linearizable, J. Math. Anal. Appl., 228 (1998), 141-156.doi: 10.1006/jmaa.1998.6122.


    B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities, J. Differential Equations, 226 (2006), 501-524.doi: 10.1016/j.jde.2005.08.016.


    B. P. Rynne, Nonresonance conditions for generalised $\phi$-Laplacian problems with jumping nonlinearities, J. Differential Equations, 247 (2009), 2364-2379.doi: 10.1016/j.jde.2009.07.012.


    K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings, J. Differential Equations, 33 (1979), 294-319.doi: 10.1016/0022-0396(79)90067-6.


    A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$-manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119-139.


    M. R. Zhang, The rotation number approach to eigenvalues of the one-dimensional $p$-Laplacian with periodic potentials, J. Lond. Math. Soc. (2), 64 (2001), 125-143.doi: 10.1017/S0024610701002277.

  • 加载中

Article Metrics

HTML views() PDF downloads(139) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint