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), 37-151. 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), 213-222.  Google Scholar

[3]

A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems," Cambridge Studies in Advanced Mathematics No. 104, Cambridge Univ. Press, London, 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), 341-356.  Google Scholar

[5]

P. A. Binding and B. P. Rynne, The spectrum of the periodic $p$-Laplacian, J. Differentiable Equations, 235 (2007), 199-218.  Google Scholar

[6]

P. A. Binding and B. P. Rynne, Variational and non-variational eigenvalues of the $p$-Laplacian, J. Differentiable Equations, 244 (2008), 24-39.  Google Scholar

[7]

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

[8]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  Google Scholar

[9]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[10]

A. Constantin, On the spectral problem for the periodic Camassa-Holm equation, J. Math. Anal. Appl., 210 (1997), 215-230.  Google Scholar

[11]

A. Constantin, A general-weighted Sturm-Liouville problem, Ann. Sci. Éc. Norm. Supér., 24 (1997), 767-782.  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), 175-211.  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), 165-186.  Google Scholar

[14]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Commun. Pure Appl. Math., 52 (1999), 949-982.  Google Scholar

[15]

M. Cuesta, Eigenvalue problems for the $p$-Laplacian wirh indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9. 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), 46-50.  Google Scholar

[17]

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468.  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), 119-123.  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), 1069-1076.  Google Scholar

[21]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538.  Google Scholar

[22]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, New-York, 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), 1-13.  Google Scholar

[24]

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Lapiacian, J. Differential Equations, 92 (1991), 226-251.  Google Scholar

[25]

L. C. Evans, "Partial Differential Equations," AMS, Rhode Island, 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), 453-464. Google Scholar

[27]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852. 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), 1185-1197.  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), 61-78. 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), 563-576.  Google Scholar

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  Google Scholar

[32]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves," Springer, Berlin, 2007, 31-49.  Google Scholar

[33]

Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differential Equations, 229 (2006), 229-256.  Google Scholar

[34]

Y. Li, Positive doubly periodic solutions of nonlinear telegraph equations, Nonlinear Anal., 55 (2003), 245-254.  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), 578-592.  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), 28-39.  Google Scholar

[37]

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

[38]

R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett., 21 (2008), 754-760.  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) 4364-4376.  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), 3379-3385.  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), 5982-5988.  Google Scholar

[42]

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

[43]

M. Montenego, Strong maximum principles for super-solutions of quasilinear elliptic equations, Nonlinear Ana1., 37 (1999) 431-448. Google Scholar

[44]

D. O'Regan and H. Wang, Positive periodic solutions of systems of second order ordinary differential equations, Positivity, 10 (2006), 285-298.  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), 487-513.  Google Scholar

[47]

P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475.  Google Scholar

[48]

B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities, J. Differential Equations, 226 (2006), 501-524.  Google Scholar

[49]

A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$-manifolds, Ann. I. H. Poincaré, Anal. non linéaire, 5 (1988), 119-139. 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), 643-662. Google Scholar

[51]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications," Vol. II/B, Berlin-Heidelberg-New York 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), 99-107.  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), 37-151. 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), 213-222.  Google Scholar

[3]

A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems," Cambridge Studies in Advanced Mathematics No. 104, Cambridge Univ. Press, London, 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), 341-356.  Google Scholar

[5]

P. A. Binding and B. P. Rynne, The spectrum of the periodic $p$-Laplacian, J. Differentiable Equations, 235 (2007), 199-218.  Google Scholar

[6]

P. A. Binding and B. P. Rynne, Variational and non-variational eigenvalues of the $p$-Laplacian, J. Differentiable Equations, 244 (2008), 24-39.  Google Scholar

[7]

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

[8]

R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664.  Google Scholar

[9]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation, Adv. Appl. Mech., 31 (1994), 1-33. Google Scholar

[10]

A. Constantin, On the spectral problem for the periodic Camassa-Holm equation, J. Math. Anal. Appl., 210 (1997), 215-230.  Google Scholar

[11]

A. Constantin, A general-weighted Sturm-Liouville problem, Ann. Sci. Éc. Norm. Supér., 24 (1997), 767-782.  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), 175-211.  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), 165-186.  Google Scholar

[14]

A. Constantin and H. P. McKean, A shallow water equation on the circle, Commun. Pure Appl. Math., 52 (1999), 949-982.  Google Scholar

[15]

M. Cuesta, Eigenvalue problems for the $p$-Laplacian wirh indefinite weights, Electron. J. Differential Equations, 33 (2001), 1-9. 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), 46-50.  Google Scholar

[17]

G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian, J. Differential Equations, 252 (2012), 2448-2468.  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), 119-123.  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), 1069-1076.  Google Scholar

[21]

E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one, Bull. London Math. Soc., 34 (2002), 533-538.  Google Scholar

[22]

K. Deimling, "Nonlinear Functional Analysis," Springer-Verlag, New-York, 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), 1-13.  Google Scholar

[24]

M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Lapiacian, J. Differential Equations, 92 (1991), 226-251.  Google Scholar

[25]

L. C. Evans, "Partial Differential Equations," AMS, Rhode Island, 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), 453-464. Google Scholar

[27]

X. L. Fan and Q. H. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problems, Nonlinear Anal., 52 (2003), 1843-1852. 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), 1185-1197.  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), 61-78. 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), 563-576.  Google Scholar

[31]

R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82.  Google Scholar

[32]

M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics, in "Tsunami and Nonlinear Waves," Springer, Berlin, 2007, 31-49.  Google Scholar

[33]

Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian, J. Differential Equations, 229 (2006), 229-256.  Google Scholar

[34]

Y. Li, Positive doubly periodic solutions of nonlinear telegraph equations, Nonlinear Anal., 55 (2003), 245-254.  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), 578-592.  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), 28-39.  Google Scholar

[37]

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

[38]

R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems, Appl. Math. Lett., 21 (2008), 754-760.  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) 4364-4376.  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), 3379-3385.  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), 5982-5988.  Google Scholar

[42]

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

[43]

M. Montenego, Strong maximum principles for super-solutions of quasilinear elliptic equations, Nonlinear Ana1., 37 (1999) 431-448. Google Scholar

[44]

D. O'Regan and H. Wang, Positive periodic solutions of systems of second order ordinary differential equations, Positivity, 10 (2006), 285-298.  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), 487-513.  Google Scholar

[47]

P. H. Rabinowitz, On bifurcation from infinity, J. Funct. Anal., 14 (1973), 462-475.  Google Scholar

[48]

B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities, J. Differential Equations, 226 (2006), 501-524.  Google Scholar

[49]

A. Szulkin, Ljusternik-Schnirelmann theory on $C^1$-manifolds, Ann. I. H. Poincaré, Anal. non linéaire, 5 (1988), 119-139. 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), 643-662. Google Scholar

[51]

E. Zeidler, "Nonlinear Functional Analysis and Its Applications," Vol. II/B, Berlin-Heidelberg-New York 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), 99-107.  Google Scholar

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