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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

Guowei Dai 1, and  Ruyun Ma 1,

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.
Keywords: Unilateral bifurcation, nodal solutions, half-quasilinear problems, $p$-Laplacian..
Mathematics Subject Classification: Primary: 34C10; Secondary: 34C2.
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

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

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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

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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

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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

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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

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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

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

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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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