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Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme
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 |
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. |
[2] |
A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems,, Comment. Math. Univ. Carolin., 31 (1990), 213.
|
[3] |
A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems,", Cambridge Studies in Advanced Mathematics No. 104, (2007).
|
[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.
|
[5] |
P. A. Binding and B. P. Rynne, The spectrum of the periodic $p$-Laplacian,, J. Differentiable Equations, 235 (2007), 199.
|
[6] |
P. A. Binding and B. P. Rynne, Variational and non-variational eigenvalues of the $p$-Laplacian,, J. Differentiable Equations, 244 (2008), 24.
|
[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.
|
[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.
|
[11] |
A. Constantin, A general-weighted Sturm-Liouville problem,, Ann. Sci. \'Ec. Norm. Sup\'er., 24 (1997), 767.
|
[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.
|
[13] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.
|
[14] |
A. Constantin and H. P. McKean, A shallow water equation on the circle,, Commun. Pure Appl. Math., 52 (1999), 949.
|
[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.
|
[17] |
G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian,, J. Differential Equations, 252 (2012), 2448.
|
[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.
|
[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. |
[20] |
E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems,, Indiana U. Math J., 23 (1974), 1069.
|
[21] |
E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. London Math. Soc., 34 (2002), 533.
|
[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.
|
[24] |
M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Lapiacian,, J. Differential Equations, 92 (1991), 226.
|
[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.
|
[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.
|
[31] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63.
|
[32] |
M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31.
|
[33] |
Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian,, J. Differential Equations, 229 (2006), 229.
|
[34] |
Y. Li, Positive doubly periodic solutions of nonlinear telegraph equations,, Nonlinear Anal., 55 (2003), 245.
|
[35] |
W. Li and X. Liu, Eigenvalue problems for second-order nonlinear dynamic equations on time scales,, J. Math. Anal. Appl., 318 (2006), 578.
|
[36] |
X. Liu and W. Li, Existence and uniqueness of positive periodic solutions of functional differential equations,, J. Math. Anal. Appl., 293 (2004), 28.
|
[37] |
J. L\'opez-Gómez, "Spectral Theory and Nonlinear Functional Analysis,", Chapman and Hall/CRC, (2001).
|
[38] |
R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems,, Appl. Math. Lett., 21 (2008), 754.
|
[39] |
R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions,, Nonlinear Anal., 71 (2009), 4364.
|
[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.
|
[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.
|
[42] |
J. Mawhin and M.Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989).
|
[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.
|
[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.
|
[47] |
P. H. Rabinowitz, On bifurcation from infinity,, J. Funct. Anal., 14 (1973), 462.
|
[48] |
B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 226 (2006), 501.
|
[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).
|
[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.
|
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. |
[2] |
A. Ambrosetti, R. M. Calahorrano and F. R. Dobarro, Global branching for discontinuous problems,, Comment. Math. Univ. Carolin., 31 (1990), 213.
|
[3] |
A. Ambrosetti and A. Malchiodi, "Nonlinear Analysis and Semilinear Elliptic Problems,", Cambridge Studies in Advanced Mathematics No. 104, (2007).
|
[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.
|
[5] |
P. A. Binding and B. P. Rynne, The spectrum of the periodic $p$-Laplacian,, J. Differentiable Equations, 235 (2007), 199.
|
[6] |
P. A. Binding and B. P. Rynne, Variational and non-variational eigenvalues of the $p$-Laplacian,, J. Differentiable Equations, 244 (2008), 24.
|
[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.
|
[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.
|
[11] |
A. Constantin, A general-weighted Sturm-Liouville problem,, Ann. Sci. \'Ec. Norm. Sup\'er., 24 (1997), 767.
|
[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.
|
[13] |
A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,, Arch. Ration. Mech. Anal., 192 (2009), 165.
|
[14] |
A. Constantin and H. P. McKean, A shallow water equation on the circle,, Commun. Pure Appl. Math., 52 (1999), 949.
|
[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.
|
[17] |
G. Dai and R. Ma, Unilateral global bifurcation phenomena and nodal solutions for $p$-Laplacian,, J. Differential Equations, 252 (2012), 2448.
|
[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.
|
[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. |
[20] |
E. N. Dancer, On the structure of solutions of non-linear eigenvalue problems,, Indiana U. Math J., 23 (1974), 1069.
|
[21] |
E. N. Dancer, Bifurcation from simple eigenvalues and eigenvalues of geometric multiplicity one,, Bull. London Math. Soc., 34 (2002), 533.
|
[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.
|
[24] |
M. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Lapiacian,, J. Differential Equations, 92 (1991), 226.
|
[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.
|
[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.
|
[31] |
R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves,, J. Fluid Mech., 455 (2002), 63.
|
[32] |
M. Lakshmanan, Integrable nonlinear wave equations and possible connections to tsunami dynamics,, in, (2007), 31.
|
[33] |
Y. H. Lee and I. Sim, Global bifurcation phenomena for singular one-dimensional $p$-Laplacian,, J. Differential Equations, 229 (2006), 229.
|
[34] |
Y. Li, Positive doubly periodic solutions of nonlinear telegraph equations,, Nonlinear Anal., 55 (2003), 245.
|
[35] |
W. Li and X. Liu, Eigenvalue problems for second-order nonlinear dynamic equations on time scales,, J. Math. Anal. Appl., 318 (2006), 578.
|
[36] |
X. Liu and W. Li, Existence and uniqueness of positive periodic solutions of functional differential equations,, J. Math. Anal. Appl., 293 (2004), 28.
|
[37] |
J. L\'opez-Gómez, "Spectral Theory and Nonlinear Functional Analysis,", Chapman and Hall/CRC, (2001).
|
[38] |
R. Ma, Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems,, Appl. Math. Lett., 21 (2008), 754.
|
[39] |
R. Ma and Y. An, Global structure of positive solutions for nonlocal boundary value problems involving integral conditions,, Nonlinear Anal., 71 (2009), 4364.
|
[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.
|
[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.
|
[42] |
J. Mawhin and M.Willem, "Critical Point Theory and Hamiltonian Systems,", Springer, (1989).
|
[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.
|
[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.
|
[47] |
P. H. Rabinowitz, On bifurcation from infinity,, J. Funct. Anal., 14 (1973), 462.
|
[48] |
B. P. Rynne, $p$-Laplacian problems with jumping nonlinearities,, J. Differential Equations, 226 (2006), 501.
|
[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).
|
[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.
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