Lyapunov inequalities for partial differential equations at radial higher eigenvalues

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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

Abstract
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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 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.

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