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Zbl 0954.35018
Kusano, Takaŝi; Jaroš, Jaroslav; Yoshida, Norio
A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 40, No.1-8, A, 381-395 (2000). ISSN 0362-546X

The well-known Picone's identity plays an important role in the study of qualitative properties of solutions of the second-order linear homogeneous differential equations. It has been recently generalized to the half-linear differential operators \align l_\alpha[y]&=(r(t)\left|y'\right|^{\alpha -1}y')'+q(t)\left|y\right|^{\alpha -1}y,\\ L_{\alpha }[y] &=(R(t)\left|z'\right|^{\alpha -1}z')'+Q(t)\left|z\right|^{\alpha -1}z, \endalign where $\alpha>0$ is a constant, and $r,q,R,Q$ are real-valued continuous functions on an interval. Using a generalization of Picone's identity to the linear elliptic operators \align p[u]&=\nabla \cdot (a(x)\nabla u)+c(x)u,\\ P[v] & =\nabla \cdot (A(x)\nabla v)+C(x)v, \endalign a number of authors developed Sturmian theory for second order linear elliptic equations. In this paper, the authors generalized Picone's identity to the half-linear partial differential operators \align p_\alpha[u]&=\nabla \cdot (a(x)\left|\nabla u\right|^{\alpha -1}\nabla u)'+c(x)\left|u\right|^{\alpha -1}u,\\ P_{\alpha }[v] & =\nabla \cdot (A(x)\left|\nabla v\right|^{\alpha -1}\nabla v)'+C(x)\left|v\right|^{\alpha-1}v,\endalign where $\alpha>0$ is a constant, and $a,c,A,C$ are continuous (continuously differentiable) functions defined in a domain $G\subset\bbfR^n$. Then the obtained Picone-type identity is applied to prove Sturmian comparison and oscillation theorems for second-order half-linear degenerate elliptic equations of the form $p_\alpha[u]=0$ or $P_\alpha[v]=0$ in an unbounded domain in $\bbfR^n$.
[Svitlana P.Rogovchenko (Mersin)]
MSC 2000:
*35B05 General behavior of solutions of PDE
35J70 Elliptic equations of degenerate type

Keywords: second-order half-linear degenerate elliptic equations; unbounded domain

Cited in: Zbl pre05142146

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