Electron. J. Diff. Equ., Vol. 2014 (2014), No. 214, pp. 1-10.

Sturm-Picone type theorems for second-order nonlinear elliptic differential equations

Aydin Tiryaki

Abstract:
The aim of this article is to give Sturm-Picone type theorems for the pair of second order nonlinear elliptic differential equations
$$\displaylines{ \hbox{div}(p_1(x)|\nabla u|^{\alpha-1}\nabla u ) +q_1(x)f_1(u)+r_1(x)g_1(u)=0,\cr \hbox{div}(p_2(x)|\nabla v|^{\alpha-1}\nabla v ) +q_2(x)f_2(v)+r_2(x)g_2(v)=0, }$$
where $|\cdot|$ denotes the Euclidean length and $\nabla= (\frac{\partial}{\partial x_1},\dots, \frac{\partial}{\partial x_{n}} )^{T}$ (the superscript T denotes the transpose). Our results include some earlier results and generalize to n-dimensions well-known comparison theorems given by Sturm, Picone and Leighton ]26.37] which play a key role in the qualitative behavior of solutions. By using generalization of n dimensional Leigton's comparison theorem, an oscillation result is given as an application.

Submitted July 16, 2014. Published October 14, 2014.
Math Subject Classifications: 35B05.
Key Words: Comparison theorem; Sturm-Picone theorem; half-linear equations, variational lemma; elliptic equations; oscillation.

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Aydin Tiryaki
Department of Mathematics and Computer Sciences
Faculty of Arts and Sciences, Izmir University
35350 Uckuyular, Izmir, Turkey
email: aydin.tiryaki@izmir.edu.tr

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