January  2013, 33(1): 111-122. doi: 10.3934/dcds.2013.33.111

Lyapunov inequalities for partial differential equations at radial higher eigenvalues

1. 

Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Granada, Campus Fuentenueva. 18071 Granada, Spain, Spain

Received  August 2011 Revised  November 2011 Published  September 2012

This paper is devoted to the study of $L_{p}$ Lyapunov-type inequalities ($ \ 1 \leq p \leq +\infty$) for linear partial differential equations at radial higher eigenvalues. More precisely, we treat the case of Neumann boundary conditions on balls in $\Bbb{R}^{N}$. It is proved that the relation between the quantities $p$ and $N/2$ plays a crucial role to obtain nontrivial and optimal Lyapunov inequalities. By using appropriate minimizing sequences and a detailed analysis about the number and distribution of zeros of radial nontrivial solutions, we show significant qualitative differences according to the studied case is subcritical, supercritical or critical.
Citation: Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 111-122. doi: 10.3934/dcds.2013.33.111
References:
[1]

A. Cañada, J. A. Montero and S. Villegas, Liapunov-type inequalities and Neumann boundary value problems at resonance, Math. Inequal. Appl., 8 (2005), 459-475.

[2]

A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal., 237 (2006), 176-193. doi: 10.1016/j.jfa.2005.12.011.

[3]

A. Cañada and S. Villegas, Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues, J. Eur. Math. Soc. (JEMS), 12 (2010), 163-178.

[4]

M. del Pino and R. F. 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.

[5]

C. L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc., 66 (1949), 289-307. doi: 10.1090/S0002-9947-1949-0032923-4.

[6]

B. Fuglede, Continuous domain dependence of the eigenvalues of the Dirichlet Laplacian and related operators in Hilbert space, J. Funct. Anal., 167 (1999), 183-200. doi: 10.1006/jfan.1999.3442.

[7]

P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[8]

Y. Li and H. Z. Wang, Neumann boundary value problems for second-order ordinary differential equations across resonance, SIAM J. Control Optim., 33 (1995), 1312-1325. doi: 10.1137/S036301299324532X.

[9]

G. Vidossich, Existence and uniqueness results for boundary value problems from the comparison of eigenvalues, ICTP Preprint Archive, 1979015, (1979).

[10]

M. Zhang, Certain classes of potentials for $p$-Laplacian to be non-degenerate, Math. Nachr., 278 (2005), 1823-1836. doi: 10.1002/mana.200410342.

show all references

References:
[1]

A. Cañada, J. A. Montero and S. Villegas, Liapunov-type inequalities and Neumann boundary value problems at resonance, Math. Inequal. Appl., 8 (2005), 459-475.

[2]

A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal., 237 (2006), 176-193. doi: 10.1016/j.jfa.2005.12.011.

[3]

A. Cañada and S. Villegas, Lyapunov inequalities for Neumann boundary conditions at higher eigenvalues, J. Eur. Math. Soc. (JEMS), 12 (2010), 163-178.

[4]

M. del Pino and R. F. 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.

[5]

C. L. Dolph, Nonlinear integral equations of the Hammerstein type, Trans. Amer. Math. Soc., 66 (1949), 289-307. doi: 10.1090/S0002-9947-1949-0032923-4.

[6]

B. Fuglede, Continuous domain dependence of the eigenvalues of the Dirichlet Laplacian and related operators in Hilbert space, J. Funct. Anal., 167 (1999), 183-200. doi: 10.1006/jfan.1999.3442.

[7]

P. Hartman, "Ordinary Differential Equations," John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[8]

Y. Li and H. Z. Wang, Neumann boundary value problems for second-order ordinary differential equations across resonance, SIAM J. Control Optim., 33 (1995), 1312-1325. doi: 10.1137/S036301299324532X.

[9]

G. Vidossich, Existence and uniqueness results for boundary value problems from the comparison of eigenvalues, ICTP Preprint Archive, 1979015, (1979).

[10]

M. Zhang, Certain classes of potentials for $p$-Laplacian to be non-degenerate, Math. Nachr., 278 (2005), 1823-1836. doi: 10.1002/mana.200410342.

[1]

Minoo Kamrani. Numerical solution of partial differential equations with stochastic Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5337-5354. doi: 10.3934/dcdsb.2019061

[2]

K. Q. Lan. Properties of kernels and eigenvalues for three point boundary value problems. Conference Publications, 2005, 2005 (Special) : 546-555. doi: 10.3934/proc.2005.2005.546

[3]

K. Q. Lan. Multiple positive eigenvalues of conjugate boundary value problems with singularities. Conference Publications, 2003, 2003 (Special) : 501-506. doi: 10.3934/proc.2003.2003.501

[4]

Antonio Cañada, Salvador Villegas. Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 877-888. doi: 10.3934/dcds.2008.20.877

[5]

Xiaofei He, X. H. Tang. Lyapunov-type inequalities for even order differential equations. Communications on Pure and Applied Analysis, 2012, 11 (2) : 465-473. doi: 10.3934/cpaa.2012.11.465

[6]

Frédéric Mazenc, Christophe Prieur. Strict Lyapunov functions for semilinear parabolic partial differential equations. Mathematical Control and Related Fields, 2011, 1 (2) : 231-250. doi: 10.3934/mcrf.2011.1.231

[7]

Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595

[8]

Paul Eloe, Jaganmohan Jonnalagadda. Quasilinearization applied to boundary value problems at resonance for Riemann-Liouville fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2719-2734. doi: 10.3934/dcdss.2020220

[9]

Zhiyuan Wen, Meirong Zhang. On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures. Discrete and Continuous Dynamical Systems - B, 2020, 25 (8) : 3257-3274. doi: 10.3934/dcdsb.2020061

[10]

Patricio Cerda, Leonelo Iturriaga, Sebastián Lorca, Pedro Ubilla. Positive radial solutions of a nonlinear boundary value problem. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1765-1783. doi: 10.3934/cpaa.2018084

[11]

Ravi P. Agarwal, Abdullah Özbekler. Lyapunov type inequalities for $n$th order forced differential equations with mixed nonlinearities. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2281-2300. doi: 10.3934/cpaa.2016037

[12]

Meina Gao, Jianjun Liu. A degenerate KAM theorem for partial differential equations with periodic boundary conditions. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5911-5928. doi: 10.3934/dcds.2020252

[13]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203

[14]

Guowei Dai, Ruyun Ma, Haiyan Wang, Feng Wang, Kuai Xu. Partial differential equations with Robin boundary condition in online social networks. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1609-1624. doi: 10.3934/dcdsb.2015.20.1609

[15]

Angelo Favini. A general approach to identification problems and applications to partial differential equations. Conference Publications, 2015, 2015 (special) : 428-435. doi: 10.3934/proc.2015.0428

[16]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967

[17]

Alberto Boscaggin, Francesca Colasuonno, Benedetta Noris. Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1921-1933. doi: 10.3934/dcdss.2020150

[18]

Cristian Bereanu, Petru Jebelean, Jean Mawhin. Radial solutions for Neumann problems with $\phi$-Laplacians and pendulum-like nonlinearities. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 637-648. doi: 10.3934/dcds.2010.28.637

[19]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[20]

Iñigo U. Erneta. Well-posedness for boundary value problems for coagulation-fragmentation equations. Kinetic and Related Models, 2020, 13 (4) : 815-835. doi: 10.3934/krm.2020028

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]