# American Institute of Mathematical Sciences

January  2015, 35(1): 99-116. doi: 10.3934/dcds.2015.35.99

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

 1 Department of Mathematics, Northwest Normal University, Lanzhou, 730070

Received  February 2013 Revised  June 2014 Published  August 2014

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.
Citation: Guowei Dai, Ruyun Ma. Unilateral global bifurcation for $p$-Laplacian with non-$p-$1-linearization nonlinearity. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 99-116. doi: 10.3934/dcds.2015.35.99
##### References:
 [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). Google Scholar [2] H. Berestycki, On some nonlinear Sturm-Liouville problems,, J. Differential Equations, 26 (1977), 375. doi: 10.1016/0022-0396(77)90086-9. Google Scholar [3] H. Brezis, Opérateurs Maximaux Monotone et Semigroup de Contractions dans les Espase de Hilbert,, Math. Studies, (1973). Google Scholar [4] G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian,, J. Differential Equations, 252 (2012), 2448. doi: 10.1016/j.jde.2011.09.026. Google Scholar [5] E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems,, Indiana U. Math J., 23 (1974), 1069. Google Scholar [6] E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. Lond. Math. Soc., 34 (2002), 533. doi: 10.1112/S002460930200108X. Google Scholar [7] M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian,, J. Differential Equations, 92 (1991), 226. doi: 10.1016/0022-0396(91)90048-E. Google Scholar [8] P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbbR^N$,, Trans. Amer. Math. Soc., 349 (1997), 171. doi: 10.1090/S0002-9947-97-01788-1. Google Scholar [9] J. Fleckinger and W. Reichel, Global solution branches for $p$-Laplacian boundary value problems,, Nonlinear Anal., 62 (2005), 53. doi: 10.1016/j.na.2003.11.015. Google Scholar [10] J. Fleckinger, R. Manásevich and Thélin, Global bifurcation from the first eigenvalue for a system of $p$-Laplacians,, Math. Nachr., 182 (1996), 217. doi: 10.1002/mana.19961820110. Google Scholar [11] J. García-Azorero and I. Peral, Existence and non-uniqueness for the $p$-Laplacian: Nonlinear eigenvalues,, Comm. Part. Diff. Equations, 12 (1987), 1389. doi: 10.1080/03605308708820534. Google Scholar [12] 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. doi: 10.1006/jdeq.2001.4031. Google Scholar [13] P. Girg and P. Takáč, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems,, Ann. Henri Poincar'e, 9 (2008), 275. doi: 10.1007/s00023-008-0356-x. Google Scholar [14] J. K. Hale, Bifurcation from simple eigenvalues for several parameter families,, Nonlinear Anal., 2 (1978), 491. doi: 10.1016/0362-546X(78)90056-1. Google Scholar [15] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function,, Comm. Partial Differential Equations, 5 (1980), 999. doi: 10.1080/03605308008820162. Google Scholar [16] 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. doi: 10.1016/j.jmaa.2004.10.054. Google Scholar [17] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1965). Google Scholar [18] 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. doi: 10.1016/S0362-546X(00)85023-3. Google Scholar [19] Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian,, J. Differential Equations, 229 (2006), 229. doi: 10.1016/j.jde.2006.03.021. Google Scholar [20] J. López-Gómez, Multiparameter local bifurcation based on the linear part,, J. Math. Anal. Appns., 138 (1989), 358. doi: 10.1016/0022-247X(89)90296-5. Google Scholar [21] J. López-Gómez, Positive periodic solutions of Lotka-Volterra Systems,, Diff. Int. Eqns., 5 (1992), 55. Google Scholar [22] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Chapman and Hall/CRC, (2001). doi: 10.1201/9781420035506. Google Scholar [23] J. López-Gómez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory,, Nonlinear Anal., 58 (2004), 749. doi: 10.1016/j.na.2004.04.011. Google Scholar [24] J. López-Gómez and C. Mora-Corral, Counting zeroes of $C^1$-Fredholm maps of index 1,, Bull. Lond. Math. Soc., 37 (2005), 778. doi: 10.1112/S0024609305004716. Google Scholar [25] J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators,, Advances in Operator Theory and Applications, (2007). Google Scholar [26] R. Ma and B. Thompson, Nodal solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 59 (2004), 707. doi: 10.1016/j.na.2004.07.030. Google Scholar [27] R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p$-Laplacian-like operators,, J. Differential Equations, 145 (1998), 367. doi: 10.1006/jdeq.1998.3425. Google Scholar [28] P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Commun. Pure Appl. Math., 23 (1970), 939. doi: 10.1002/cpa.3160230606. Google Scholar [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar [30] P. H. Rabinowitz, On bifurcation from infinity,, J. Funct. Anal., 14 (1973), 462. doi: 10.1016/0022-0396(73)90061-2. Google Scholar [31] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems,, Rocky Mountain J. Math., 3 (1973), 161. doi: 10.1216/RMJ-1973-3-2-161. Google Scholar [32] B. P. Rynne, Bifurcation from zero or infinity in Sturm-Liouville problems which are not linearizable,, J. Math. Anal. Appl., 228 (1998), 141. doi: 10.1006/jmaa.1998.6122. Google Scholar [33] B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 226 (2006), 501. doi: 10.1016/j.jde.2005.08.016. Google Scholar [34] B. P. Rynne, Nonresonance conditions for generalised $\phi$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 247 (2009), 2364. doi: 10.1016/j.jde.2009.07.012. Google Scholar [35] K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings,, J. Differential Equations, 33 (1979), 294. doi: 10.1016/0022-0396(79)90067-6. Google Scholar [36] A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$-manifolds,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119. Google Scholar [37] 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. doi: 10.1017/S0024610701002277. Google Scholar

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##### References:
 [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). Google Scholar [2] H. Berestycki, On some nonlinear Sturm-Liouville problems,, J. Differential Equations, 26 (1977), 375. doi: 10.1016/0022-0396(77)90086-9. Google Scholar [3] H. Brezis, Opérateurs Maximaux Monotone et Semigroup de Contractions dans les Espase de Hilbert,, Math. Studies, (1973). Google Scholar [4] G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian,, J. Differential Equations, 252 (2012), 2448. doi: 10.1016/j.jde.2011.09.026. Google Scholar [5] E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems,, Indiana U. Math J., 23 (1974), 1069. Google Scholar [6] E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. Lond. Math. Soc., 34 (2002), 533. doi: 10.1112/S002460930200108X. Google Scholar [7] M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian,, J. Differential Equations, 92 (1991), 226. doi: 10.1016/0022-0396(91)90048-E. Google Scholar [8] P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbbR^N$,, Trans. Amer. Math. Soc., 349 (1997), 171. doi: 10.1090/S0002-9947-97-01788-1. Google Scholar [9] J. Fleckinger and W. Reichel, Global solution branches for $p$-Laplacian boundary value problems,, Nonlinear Anal., 62 (2005), 53. doi: 10.1016/j.na.2003.11.015. Google Scholar [10] J. Fleckinger, R. Manásevich and Thélin, Global bifurcation from the first eigenvalue for a system of $p$-Laplacians,, Math. Nachr., 182 (1996), 217. doi: 10.1002/mana.19961820110. Google Scholar [11] J. García-Azorero and I. Peral, Existence and non-uniqueness for the $p$-Laplacian: Nonlinear eigenvalues,, Comm. Part. Diff. Equations, 12 (1987), 1389. doi: 10.1080/03605308708820534. Google Scholar [12] 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. doi: 10.1006/jdeq.2001.4031. Google Scholar [13] P. Girg and P. Takáč, Bifurcations of positive and negative continua in quasilinear elliptic eigenvalue problems,, Ann. Henri Poincar'e, 9 (2008), 275. doi: 10.1007/s00023-008-0356-x. Google Scholar [14] J. K. Hale, Bifurcation from simple eigenvalues for several parameter families,, Nonlinear Anal., 2 (1978), 491. doi: 10.1016/0362-546X(78)90056-1. Google Scholar [15] P. Hess and T. Kato, On some linear and nonlinear eigenvalue problems with an indefinite weight function,, Comm. Partial Differential Equations, 5 (1980), 999. doi: 10.1080/03605308008820162. Google Scholar [16] 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. doi: 10.1016/j.jmaa.2004.10.054. Google Scholar [17] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Macmillan, (1965). Google Scholar [18] 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. doi: 10.1016/S0362-546X(00)85023-3. Google Scholar [19] Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian,, J. Differential Equations, 229 (2006), 229. doi: 10.1016/j.jde.2006.03.021. Google Scholar [20] J. López-Gómez, Multiparameter local bifurcation based on the linear part,, J. Math. Anal. Appns., 138 (1989), 358. doi: 10.1016/0022-247X(89)90296-5. Google Scholar [21] J. López-Gómez, Positive periodic solutions of Lotka-Volterra Systems,, Diff. Int. Eqns., 5 (1992), 55. Google Scholar [22] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis,, Chapman and Hall/CRC, (2001). doi: 10.1201/9781420035506. Google Scholar [23] J. López-Gómez and C. Mora-Corral, Minimal complexity of semi-bounded components in bifurcation theory,, Nonlinear Anal., 58 (2004), 749. doi: 10.1016/j.na.2004.04.011. Google Scholar [24] J. López-Gómez and C. Mora-Corral, Counting zeroes of $C^1$-Fredholm maps of index 1,, Bull. Lond. Math. Soc., 37 (2005), 778. doi: 10.1112/S0024609305004716. Google Scholar [25] J. López-Gómez and C. Mora-Corral, Algebraic Multiplicity of Eigenvalues of Linear Operators,, Advances in Operator Theory and Applications, (2007). Google Scholar [26] R. Ma and B. Thompson, Nodal solutions for nonlinear eigenvalue problems,, Nonlinear Anal., 59 (2004), 707. doi: 10.1016/j.na.2004.07.030. Google Scholar [27] R. Manásevich and J. Mawhin, Periodic solutions for nonlinear systems with $p$-Laplacian-like operators,, J. Differential Equations, 145 (1998), 367. doi: 10.1006/jdeq.1998.3425. Google Scholar [28] P. H. Rabinowitz, Nonlinear Sturm-Liouville problems for second order ordinary differential equations,, Commun. Pure Appl. Math., 23 (1970), 939. doi: 10.1002/cpa.3160230606. Google Scholar [29] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487. doi: 10.1016/0022-1236(71)90030-9. Google Scholar [30] P. H. Rabinowitz, On bifurcation from infinity,, J. Funct. Anal., 14 (1973), 462. doi: 10.1016/0022-0396(73)90061-2. Google Scholar [31] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems,, Rocky Mountain J. Math., 3 (1973), 161. doi: 10.1216/RMJ-1973-3-2-161. Google Scholar [32] B. P. Rynne, Bifurcation from zero or infinity in Sturm-Liouville problems which are not linearizable,, J. Math. Anal. Appl., 228 (1998), 141. doi: 10.1006/jmaa.1998.6122. Google Scholar [33] B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 226 (2006), 501. doi: 10.1016/j.jde.2005.08.016. Google Scholar [34] B. P. Rynne, Nonresonance conditions for generalised $\phi$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 247 (2009), 2364. doi: 10.1016/j.jde.2009.07.012. Google Scholar [35] K. Schmitt and H. L. Smith, On eigenvalue problems for nondifferentiable mappings,, J. Differential Equations, 33 (1979), 294. doi: 10.1016/0022-0396(79)90067-6. Google Scholar [36] A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$-manifolds,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 119. Google Scholar [37] 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. doi: 10.1017/S0024610701002277. Google Scholar
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