Advanced Search
Article Contents
Article Contents

Bifurcation and one-sign solutions of the $p$-Laplacian involving a nonlinearity with zeros

Abstract Related Papers Cited by
  • In this paper, we use bifurcation method to investigate the existence and multiplicity of one-sign solutions of the $p$-Laplacian involving a linear/superlinear nonlinearity with zeros. To do this, we first establish a bifurcation theorem from infinity for nonlinear operator equation with homogeneous operator. To deal with the superlinear case, we establish several topological results involving superior limit.
    Mathematics Subject Classification: Primary: 47J15, 47J05; Secondary: 35J60, 35B40.


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

    A. Anane, Simplicité et isolation de la première valeur propre du $p$-Laplacien avec poids, Comptes Rendus Acad. Sc. Paris, Série I, 305 (1987), 725-728.


    H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rew., 18 (1976), 620-709.doi: 10.1137/1018114.


    A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.doi: 10.1006/jfan.1994.1078.


    A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems, Comment. Math. Univ. Carolin., 31 (1990), 213-222.


    A. Ambrosetti and P. Hess, Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73 (1980), 411-422.doi: 10.1016/0022-247X(80)90287-5.


    A. Ambrosetti, J. G. Azorero and I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal., 137 (1996), 219-242.doi: 10.1006/jfan.1996.0045.


    A. Ambrosetti and A. Malchiodi, Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics No. 104, Cambridge Univ. Press, 2007.doi: 10.1017/CBO9780511618260.


    C. J. Amick and R. E. L. Turner, A global branch of steady vortex rings, J. Rein. Angew. Math., 384 (1988), 1-23.


    D. Arcoya, J. I. Diaz and L. Tello, $S$-shaped bifurcation branch in a quasilinear multivalued model arising in climatoloty, J. Differential Equations, 150 (1998), 215-225.doi: 10.1006/jdeq.1998.3502.


    D. Arcoya and J. L. Gámez, Bifurcation theory and related problems: anti-maximum principle and resonance, Comm. Partial Differential Equations, 26 (2001), 1879-1911.doi: 10.1081/PDE-100107462.


    M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, 1977.


    A. Cañada, P. Drábek and J. L. Gámez, Existence of positive solutions for some problems with nonlinear diffusion, Trans. Amer. Math. Soc., 349 (1997), 4231-4249.doi: 10.1090/S0002-9947-97-01947-8.


    G. Dai, Global branching for discontinuous problems involving the $p$-Laplacian, Electron. J. Differential Equations, 44 (2013), 1-10.


    G. Dai, Eigenvalue, global bifurcation and positive solutions for a class of nonlocal elliptic equations, Topol. Methods Nonlinear Anal., Inpress.


    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.


    G. Dai, R. Ma and Y. Lu, Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition, J. Math. Anal. Appl., 397 (2013), 119-123.doi: 10.1016/j.jmaa.2012.07.056.


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


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


    K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, New-York, 1985.doi: 10.1007/978-3-662-00547-7.


    M. Delgado and A. Suárez, On the existence and multiplicity of positive solutions for some indefinite nonlinear eigenvalue problem, Proc. Amer. Math. Soc., 132 (2004), 1721-1728.doi: 10.1090/S0002-9939-04-07233-8.


    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.


    X. L. Fan, Global $C^{1,\alpha}$ regularity for variable exponent elliptic equations in divergence form, J. Differential Equations, 235 (2007), 397-417.doi: 10.1016/j.jde.2007.01.008.


    X. L. Fan, On the sub-supersolution method for $p(x)$-Laplacian equations, J. Math. Anal. Appl., 330 (2007), 665-682.doi: 10.1016/j.jmaa.2006.07.093.


    X. L. Fan and D. Zhao, A class of De Giorgi type and Hölder continuity, Nonlinear Anal., 36 (1999), 295-318.doi: 10.1016/S0362-546X(97)00628-7.


    D. G. De Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal., 199 (2003), 452-467.doi: 10.1016/S0022-1236(02)00060-5.


    D. G. de Figueiredo, P. L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61 (1982), 41-63.


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


    P. Girg and P. Takáč, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (2008), 275-327.doi: 10.1007/s00023-008-0356-x.


    M. Guedda and L. Véron, Bifurcation phenomena associated to the $p$-Laplace operator, Trans. Amer. Math. Soc., 310 (1988), 419-431.doi: 10.2307/2001132.


    K. C. Hung and S. H. Wang, A complete classification of bifurcation diagrams of classes of multiparameter $p$-Laplacian boundary value problems, J. Differential Equations, 246 (2009), 1568-1599.doi: 10.1016/j.jde.2008.10.035.


    L. Iturriaga et al., Positive solutions of the $p$-Laplacian involving a superlinear nonlinearity with zeros, J. Differential Equations, 248 (2010), 309-327.doi: 10.1016/j.jde.2009.08.008.


    H. Kielhöfer, Bifurcation Theory: An Introduction with Applications to PDEs, Springer, New York, 2004.doi: 10.1007/b97365.


    M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New York, 1964.


    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.


    G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.doi: 10.1016/0362-546X(88)90053-3.


    P. L. Lions, On the existence of positive solutions of semilinear elliptic equations, SIAM Rev., 24 (1982), 441-467.doi: 10.1137/1024101.


    P. Liu, J. Shi and Y. Wang, Imperfect transcritical and pitchfork bifurcations, J. Funct. Anal., 251 (2007), 573-600.doi: 10.1016/j.jfa.2007.06.015.


    P. Liu, J. Shi and Y. Wang, Bifurcation from a degenerate simple eigenvalue, J. Funct. Anal., 264 (2013), 2269-2299.doi: 10.1016/j.jfa.2013.02.010.


    Z. Liu, Positive solutions of superlinear elliptic equations, J. Funct. Anal., 167 (1999), 370-398.doi: 10.1006/jfan.1999.3446.


    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, Algebraic Multiplicity of Eigenvalues of Linear Operators, Advances in Operator Theory and Applications Vol. 177, Birkhaüser, Basel, 2007.


    R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett., 21 (2008), 754-760.doi: 10.1016/j.aml.2007.07.029.


    R. Ma and G. Dai, Global bifurcation and nodal solutions for a Sturm-Liouville problem with a nonsmooth nonlineariy, J. Funct. Anal., 265 (2013), 1443-1459.doi: 10.1016/j.jfa.2013.06.017.


    I. Peral, Multiplicity of solutions for the $p$-Laplacian, ICTP SMR 990/1, 1997.


    S. Prashanth and K. Sreenadh, Multiplicity results in a ball for $p$-Laplace equation with positive nonlinearity, Adv. Differential Equations, 7 (2002), 877-896.


    P. Pucci and J. Serrin, The strong maximum principle revisited, J. Differential Equations, 196 (2004), 1-66.doi: 10.1016/j.jde.2003.05.001.


    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.


    J. Shi, Persistence and bifurcation of degenerate solutions, J. Funct. Anal., 169 (1999), 494-531.doi: 10.1006/jfan.1999.3483.


    J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812.doi: 10.1016/j.jde.2008.09.009.


    P. Takáč, L. Tello and M. Ulm, Variational problems with a $p$-homogeneous energy, Positivity, 6 (2002), 75-94.doi: 10.1023/A:1012088127719.


    P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.doi: 10.1016/0022-0396(84)90105-0.


    M. Väth, Global bifurcation of the $p$-Laplacian and related operators, J. Differential Equations, 213 (2005), 389-409.doi: 10.1016/j.jde.2004.10.005.


    G. T. Whyburn, Topological Analysis, Princeton University Press, Princeton, 1964.

  • 加载中

Article Metrics

HTML views() PDF downloads(189) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint