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, 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), 11 pp.  Google Scholar

[2]

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

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

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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.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbbR^N$, Trans. Amer. Math. Soc., 349 (1997), 171-188. 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), 53-70. 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), 217-242. doi: 10.1002/mana.19961820110.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  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), 999-1030. 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), 61-78. 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, New York, 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), 381-395. doi: 10.1016/S0362-546X(00)85023-3.  Google Scholar

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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.  Google Scholar

[20]

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.  Google Scholar

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

[22]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 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), 749-777. 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), 778-792. 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, Vol. 177, Birkhaüser, Basel, 2007.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

[30]

P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475. 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), 161-202. 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), 141-156. doi: 10.1006/jmaa.1998.6122.  Google Scholar

[33]

B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities, J. Differential Equations, 226 (2006), 501-524. 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), 2364-2379. 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), 294-319. 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), 119-139.  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), 125-143. doi: 10.1017/S0024610701002277.  Google Scholar

show all references

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), 11 pp.  Google Scholar

[2]

H. Berestycki, On some nonlinear Sturm-Liouville problems, J. Differential Equations, 26 (1977), 375-390. 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, 5, North-Holland, Amsterdam, 1973.  Google Scholar

[4]

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468. 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), 1069-1076.  Google Scholar

[6]

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.  Google Scholar

[7]

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.  Google Scholar

[8]

P. Drábek and Y. X. Huang, Bifurcation problems for the $p$-Laplacian in $\mathbbR^N$, Trans. Amer. Math. Soc., 349 (1997), 171-188. 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), 53-70. 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), 217-242. 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), 1389-1430. 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), 27-43. 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), 275-327. 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), 491-497. 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), 999-1030. 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), 61-78. 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, New York, 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), 381-395. 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), 229-256. 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), 358-370. 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), 55-72.  Google Scholar

[22]

J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman and Hall/CRC, Boca Raton, 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), 749-777. 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), 778-792. 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, Vol. 177, Birkhaüser, Basel, 2007.  Google Scholar

[26]

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.  Google Scholar

[27]

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.  Google Scholar

[28]

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.  Google Scholar

[29]

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.  Google Scholar

[30]

P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475. 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), 161-202. 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), 141-156. doi: 10.1006/jmaa.1998.6122.  Google Scholar

[33]

B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities, J. Differential Equations, 226 (2006), 501-524. 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), 2364-2379. 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), 294-319. 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), 119-139.  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), 125-143. doi: 10.1017/S0024610701002277.  Google Scholar

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