November  2013, 12(6): 2839-2872. doi: 10.3934/cpaa.2013.12.2839

Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian

1. 

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

2. 

Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100

Received  December 2012 Revised  February 2013 Published  May 2013

In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight. By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight. In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$. Furthermore, under some natural hypotheses on perturbation function, we show that $(\lambda_0^\nu,0)$ is a bifurcation point of the above problems and there are two distinct unbounded sub-continua $C_\nu^{+}$ and $C_\nu^{-}$, consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$, where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions for a class of quasilinear periodic boundary problems with the sign-changing weight. Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are also studied.
Citation: Guowei Dai, Ruyun Ma, Haiyan Wang. Eigenvalues, bifurcation and one-sign solutions for the periodic $p$-Laplacian. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2839-2872. doi: 10.3934/cpaa.2013.12.2839
References:
[1]

M. S. Alber, R. Camassa, D. Holm and J. E. Marsden, The geometry of peaked solitons of a class of integrable PDE's,, Lett. Math. Phys., 32 (1994), 37. doi: 10.2307/2152750. Google Scholar

[2]

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

[3]

A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems,", Cambridge Studies in Advanced Mathematics No. 104, (2007). Google Scholar

[4]

F. M. Atici and G. S. Guseinov, On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions,, J. Comput. Appl. Math., 132 (2001), 341. Google Scholar

[5]

P. A. Binding and B. P. Rynne, The spectrum of the periodic $p$-Laplacian,, J. Differentiable Equations, 235 (2007), 199. Google Scholar

[6]

P. A. Binding and B. P. Rynne, Variational and non-variational eigenvalues of the $p$-Laplacian,, J. Differentiable Equations, 244 (2008), 24. Google Scholar

[7]

H. Brezis, "Analyse Fonctioneile. Theéorie et Applications,", Masson, (1983). Google Scholar

[8]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. Google Scholar

[9]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. Google Scholar

[10]

A. Constantin, On the spectral problem for the periodic Camassa-Holm equation,, J. Math. Anal. Appl., 210 (1997), 215. Google Scholar

[11]

A. Constantin, A general-weighted Sturm-Liouville problem,, Ann. Sci. \'Ec. Norm. Sup\'er., 24 (1997), 767. Google Scholar

[12]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,, Fluid Dynam. Res., 40 (2008), 175. Google Scholar

[13]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. Google Scholar

[14]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Commun. Pure Appl. Math., 52 (1999), 949. Google Scholar

[15]

M. Cuesta, Eigenvalue problems for the $p$-Laplacian wirh indefinite weights,, Electron. J. Differential Equations, 33 (2001), 1. Google Scholar

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G. Dai, Bifurcation and nodal solutions for $p$-Laplacian problems with non-asymptotic nonlinearity at 0 or $\infty$,, Appl. Math. Lett., 26 (2013), 46. Google Scholar

[17]

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

[18]

G. Dai, R. Ma and Y. Lu, Bifurcation from infinity and nodal solutions of quasilinear problems without signum condition,, J. Math. Anal. Appl., 397 (2013), 119. Google Scholar

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G. Dai, et al., Global bifurcation and nodal solutions of $N$-dimensional $p$-Laplacian in unit ball,, Appl. Anal., (2013). doi: 10.1080/00036811.2012.678333. Google Scholar

[20]

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

[21]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. London Math. Soc., 34 (2002), 533. Google Scholar

[22]

K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1987). Google Scholar

[23]

M. Del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')'+f(t,u)=0$, $u(0)=u(T)=0$, $p>1$,, J. Differential Equations, 80 (1989), 1. Google Scholar

[24]

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Lapiacian,, J. Differential Equations, 92 (1991), 226. Google Scholar

[25]

L. C. Evans, "Partial Differential Equations,", AMS, (1998). Google Scholar

[26]

X. L. Fan and X. Fan, A Knobloch-type result for $p(t)$-Laplacian systems,, J. Math. Anal. Appl., 282 (2003), 453. Google Scholar

[27]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problems,, Nonlinear Anal., 52 (2003), 1843. Google Scholar

[28]

J. R. Graef, L. Kong and H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem,, J. Differential Equations, 245 (2008), 1185. Google Scholar

[29]

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. Google Scholar

[30]

D. Jiang et. al., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces,, J. Math. Anal. Appl., 286 (2003), 563. Google Scholar

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. Google Scholar

[32]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31. Google Scholar

[33]

Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian,, J. Differential Equations, 229 (2006), 229. Google Scholar

[34]

Y. Li, Positive doubly periodic solutions of nonlinear telegraph equations,, Nonlinear Anal., 55 (2003), 245. Google Scholar

[35]

W. Li and X. Liu, Eigenvalue problems for second-order nonlinear dynamic equations on time scales,, J. Math. Anal. Appl., 318 (2006), 578. Google Scholar

[36]

X. Liu and W. Li, Existence and uniqueness of positive periodic solutions of functional differential equations,, J. Math. Anal. Appl., 293 (2004), 28. Google Scholar

[37]

J. L\'opez-Gómez, "Spectral Theory and Nonlinear Functional Analysis,", Chapman and Hall/CRC, (2001). Google Scholar

[38]

R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems,, Appl. Math. Lett., 21 (2008), 754. Google Scholar

[39]

R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions,, Nonlinear Anal., 71 (2009), 4364. Google Scholar

[40]

R. Ma, J. Xu and X. Han, Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight,, Nonlinear Anal., 74 (2011), 3379. Google Scholar

[41]

R. Ma, J. Xu and X. Han, Global structure of positive solutions for superlinear second-order periodic boundary value problems,, Appl. Math. Comput., 218 (2012), 5982. Google Scholar

[42]

J. Mawhin and M.Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989). Google Scholar

[43]

M. Montenego, Strong maximum principles for super-solutions of quasilinear elliptic equations,, Nonlinear Ana1., 37 (1999), 431. Google Scholar

[44]

D. O'Regan and H. Wang, Positive periodic solutions of systems of second order ordinary differential equations,, Positivity, 10 (2006), 285. Google Scholar

[45]

I. Peral, "Multiplicity of Solutions for the $p$-Laplacian,", ICTP SMR 990/1, (1997). Google Scholar

[46]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487. Google Scholar

[47]

P. H. Rabinowitz, On bifurcation from infinity,, J. Funct. Anal., 14 (1973), 462. Google Scholar

[48]

B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 226 (2006), 501. Google Scholar

[49]

A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$-manifolds,, Ann. I. H. Poincar\'e, 5 (1988), 119. Google Scholar

[50]

P. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnosel'skii fixed point theorem,, J. Differential Equations, 190 (2003), 643. Google Scholar

[51]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications,", Vol. II/B, (1985). Google Scholar

[52]

Z. Zhang and J. Wang, On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,, J. Math. Anal. Appl., 281 (2003), 99. Google Scholar

show all references

References:
[1]

M. S. Alber, R. Camassa, D. Holm and J. E. Marsden, The geometry of peaked solitons of a class of integrable PDE's,, Lett. Math. Phys., 32 (1994), 37. doi: 10.2307/2152750. Google Scholar

[2]

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

[3]

A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems,", Cambridge Studies in Advanced Mathematics No. 104, (2007). Google Scholar

[4]

F. M. Atici and G. S. Guseinov, On the existence of positive solutions for nonlinear differential equations with periodic boundary conditions,, J. Comput. Appl. Math., 132 (2001), 341. Google Scholar

[5]

P. A. Binding and B. P. Rynne, The spectrum of the periodic $p$-Laplacian,, J. Differentiable Equations, 235 (2007), 199. Google Scholar

[6]

P. A. Binding and B. P. Rynne, Variational and non-variational eigenvalues of the $p$-Laplacian,, J. Differentiable Equations, 244 (2008), 24. Google Scholar

[7]

H. Brezis, "Analyse Fonctioneile. Theéorie et Applications,", Masson, (1983). Google Scholar

[8]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661. Google Scholar

[9]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Mech., 31 (1994), 1. Google Scholar

[10]

A. Constantin, On the spectral problem for the periodic Camassa-Holm equation,, J. Math. Anal. Appl., 210 (1997), 215. Google Scholar

[11]

A. Constantin, A general-weighted Sturm-Liouville problem,, Ann. Sci. \'Ec. Norm. Sup\'er., 24 (1997), 767. Google Scholar

[12]

A. Constantin and R. S. Johnson, Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis,, Fluid Dynam. Res., 40 (2008), 175. Google Scholar

[13]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165. Google Scholar

[14]

A. Constantin and H. P. McKean, A shallow water equation on the circle,, Commun. Pure Appl. Math., 52 (1999), 949. Google Scholar

[15]

M. Cuesta, Eigenvalue problems for the $p$-Laplacian wirh indefinite weights,, Electron. J. Differential Equations, 33 (2001), 1. Google Scholar

[16]

G. Dai, Bifurcation and nodal solutions for $p$-Laplacian problems with non-asymptotic nonlinearity at 0 or $\infty$,, Appl. Math. Lett., 26 (2013), 46. Google Scholar

[17]

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

[18]

G. Dai, R. Ma and Y. Lu, Bifurcation from infinity and nodal solutions of quasilinear problems without signum condition,, J. Math. Anal. Appl., 397 (2013), 119. Google Scholar

[19]

G. Dai, et al., Global bifurcation and nodal solutions of $N$-dimensional $p$-Laplacian in unit ball,, Appl. Anal., (2013). doi: 10.1080/00036811.2012.678333. Google Scholar

[20]

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

[21]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. London Math. Soc., 34 (2002), 533. Google Scholar

[22]

K. Deimling, "Nonlinear Functional Analysis,", Springer-Verlag, (1987). Google Scholar

[23]

M. Del Pino, M. Elgueta and R. Manásevich, A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')'+f(t,u)=0$, $u(0)=u(T)=0$, $p>1$,, J. Differential Equations, 80 (1989), 1. Google Scholar

[24]

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Lapiacian,, J. Differential Equations, 92 (1991), 226. Google Scholar

[25]

L. C. Evans, "Partial Differential Equations,", AMS, (1998). Google Scholar

[26]

X. L. Fan and X. Fan, A Knobloch-type result for $p(t)$-Laplacian systems,, J. Math. Anal. Appl., 282 (2003), 453. Google Scholar

[27]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problems,, Nonlinear Anal., 52 (2003), 1843. Google Scholar

[28]

J. R. Graef, L. Kong and H. Wang, Existence, multiplicity, and dependence on a parameter for a periodic boundary value problem,, J. Differential Equations, 245 (2008), 1185. Google Scholar

[29]

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. Google Scholar

[30]

D. Jiang et. al., Multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces,, J. Math. Anal. Appl., 286 (2003), 563. Google Scholar

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63. Google Scholar

[32]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31. Google Scholar

[33]

Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian,, J. Differential Equations, 229 (2006), 229. Google Scholar

[34]

Y. Li, Positive doubly periodic solutions of nonlinear telegraph equations,, Nonlinear Anal., 55 (2003), 245. Google Scholar

[35]

W. Li and X. Liu, Eigenvalue problems for second-order nonlinear dynamic equations on time scales,, J. Math. Anal. Appl., 318 (2006), 578. Google Scholar

[36]

X. Liu and W. Li, Existence and uniqueness of positive periodic solutions of functional differential equations,, J. Math. Anal. Appl., 293 (2004), 28. Google Scholar

[37]

J. L\'opez-Gómez, "Spectral Theory and Nonlinear Functional Analysis,", Chapman and Hall/CRC, (2001). Google Scholar

[38]

R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems,, Appl. Math. Lett., 21 (2008), 754. Google Scholar

[39]

R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions,, Nonlinear Anal., 71 (2009), 4364. Google Scholar

[40]

R. Ma, J. Xu and X. Han, Global bifurcation of positive solutions of a second-order periodic boundary value problem with indefinite weight,, Nonlinear Anal., 74 (2011), 3379. Google Scholar

[41]

R. Ma, J. Xu and X. Han, Global structure of positive solutions for superlinear second-order periodic boundary value problems,, Appl. Math. Comput., 218 (2012), 5982. Google Scholar

[42]

J. Mawhin and M.Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989). Google Scholar

[43]

M. Montenego, Strong maximum principles for super-solutions of quasilinear elliptic equations,, Nonlinear Ana1., 37 (1999), 431. Google Scholar

[44]

D. O'Regan and H. Wang, Positive periodic solutions of systems of second order ordinary differential equations,, Positivity, 10 (2006), 285. Google Scholar

[45]

I. Peral, "Multiplicity of Solutions for the $p$-Laplacian,", ICTP SMR 990/1, (1997). Google Scholar

[46]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems,, J. Funct. Anal., 7 (1971), 487. Google Scholar

[47]

P. H. Rabinowitz, On bifurcation from infinity,, J. Funct. Anal., 14 (1973), 462. Google Scholar

[48]

B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 226 (2006), 501. Google Scholar

[49]

A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$-manifolds,, Ann. I. H. Poincar\'e, 5 (1988), 119. Google Scholar

[50]

P. Torres, Existence of one-signed periodic solutions of some second-order differential equations via a Krasnosel'skii fixed point theorem,, J. Differential Equations, 190 (2003), 643. Google Scholar

[51]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications,", Vol. II/B, (1985). Google Scholar

[52]

Z. Zhang and J. Wang, On existence and multiplicity of positive solutions to periodic boundary value problems for singular nonlinear second order differential equations,, J. Math. Anal. Appl., 281 (2003), 99. Google Scholar

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