Lower bounds for generalized eigenvalues of the quasilinear systems

Abstract In this paper, we establish several new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which generalize or improve all related existing ones. Then we use the Lyapunov-type inequalities obtained in this paper to derive a better lower bound for the generalized eigenvalues of the one-dimensional quasilinear elliptic system with the Dirichlet boundary conditions.

[1]  Julián Fernández Bonder,et al.  Estimates for eigenvalues of quasilinear elliptic systems. Part II , 2008 .

[2]  Aurel Wintner,et al.  On an Oscillation Criterion of Liapounoff , 1951 .

[3]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[4]  Laurent Veron,et al.  Bifurcation phenomena associated to the p-Laplace operator , 1988 .

[5]  Xiaojing Yang,et al.  On inequalities of Lyapunov type , 2003, Appl. Math. Comput..

[6]  W. Walter,et al.  Sturm-Liouville theory for the radial p -operator , 1998 .

[7]  Aydin Tiryaki,et al.  On Lyapunov-type inequality for quasilinear systems , 2010, Appl. Math. Comput..

[8]  T. Bhattacharya.  A proof of the Faber-Krahn inequality for the first eigenvalue of thep-Laplacian , 1999 .

[9]  Julián Fernández Bonder,et al.  Precise asymptotic of eigenvalues of resonant quasilinear systems , 2008, 0811.1542.

[10]  Juan Pablo Pinasco,et al.  Estimates for eigenvalues of quasilinear elliptic systems , 2006 .

[11]  Evans M. Harrell.  Geometric lower bounds for the spectrum of elliptic PDEs with Dirichlet conditions in part , 2004 .

[12]  Juan Pablo Pinasco.  Lower bounds for eigenvalues of the one-dimensional $p$-Laplacian , 2004 .

[13]  A. M. Li︠a︡punov.  Problème général de la stabilité du mouvement , 1949 .

[14]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[15]  Pavel Drábek,et al.  The Fredholm alternative at the first eigenvalue for the one-dimensional p-Laplacian , 1998 .

[16]  On the asymptotic behavior of eigenvalues of the radial p-laplacian , 2005 .

[17]  Ondřej Došlý,et al.  Half-linear differential equations , 2005 .

[18]  Ravi P. Agarwal,et al.  Lyapunov and Wirtinger inequalities , 2004, Appl. Math. Lett..

[19]  Zeev Nehari.  Some eigenvalue estimates , 1959 .

[20]  Pavel Drábek,et al.  On the closed solution to some nonhomogeneous eigenvalue problems with $p$-Laplacian , 1999, Differential and Integral Equations.

[21]  Mauro de Lima Santos,et al.  Decay rates for solutions of semilinear wave equations with a memory condition at the boundary. , 2002 .

[22]  Devrim Çakmak,et al.  Lyapunov-type inequality for a class of Dirichlet quasilinear systems involving the (p1,p2,…,pn)-Laplacian , 2010 .

[23]  Wolfgang Walter,et al.  Sturm-Liouville theory for the radial $\Delta_p$-operator , 1998 .

[24]  Juan Pablo Pinasco.  Comparison of eigenvalues for the p-Laplacian with integral inequalities , 2006, Appl. Math. Comput..

[25]  J. P. Pinasco.  The Asymptotic Behavior of Nonlinear Eigenvalues , 2007 .

[26]  O. Chakrone,et al.  Spectrum of one dimensional p-Laplacian operator with indefinite weight , 2002 .

[27]  M. Belloni,et al.  Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators , 2003 .