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

Lyapunov inequalities for partial differential equations at radial higher eigenvalues

Antonio Cañada 1, and  Salvador Villegas 1,

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.
Keywords: Neumann boundary value problems, Lyapunov inequalities, radial eigenvalues., partial differential equations.
Mathematics Subject Classification: Primary: 35J25, 35B07; Secondary: 35J2.
Citation: Antonio Cañada, Salvador Villegas. Lyapunov inequalities for partial differential equations at radial higher eigenvalues. Discrete & 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

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