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

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.

Keywords

• Comparison theorem
• Sturm-Picone theorem
• half-linear equations,
• variational lemma
• elliptic equations
• oscillation