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 & Continuous Dynamical Systems - A, 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.   Google Scholar

[2]

A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations,, J. Funct. Anal., 237 (2006), 176.  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.   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.  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.  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.  doi: 10.1006/jfan.1999.3442.  Google Scholar

[7]

P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (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.  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, (1979).   Google Scholar

[10]

M. Zhang, Certain classes of potentials for $p$-Laplacian to be non-degenerate,, Math. Nachr., 278 (2005), 1823.  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.   Google Scholar

[2]

A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations,, J. Funct. Anal., 237 (2006), 176.  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.   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.  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.  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.  doi: 10.1006/jfan.1999.3442.  Google Scholar

[7]

P. Hartman, "Ordinary Differential Equations,", John Wiley & Sons, (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.  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, (1979).   Google Scholar

[10]

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

[1]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[2]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[3]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[4]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[5]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[6]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[7]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[8]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[9]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[10]

Anna Canale, Francesco Pappalardo, Ciro Tarantino. Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020274

[11]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[12]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[13]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[14]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 16: 331-348. doi: 10.3934/jmd.2020012

[15]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020271

[16]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[17]

Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020436

[18]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[19]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[20]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]