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A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order. (English) Zbl 0954.35018
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 \[ \begin{aligned} l_\alpha[y]&=(r(t)\left|y'\right|^{\alpha -1}y')'+q(t)\left|y\right|^{\alpha -1}y,\\ L_{\alpha }[y] &=(R(t)\left|z'\right|^{\alpha -1}z')'+Q(t)\left|z\right|^{\alpha -1}z, \end{aligned} \] 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 \[ \begin{aligned} p[u]&=\nabla \cdot (a(x)\nabla u)+c(x)u,\\ P[v] & =\nabla \cdot (A(x)\nabla v)+C(x)v, \end{aligned} \] 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 \[ \begin{aligned} p_\alpha[u]&=\nabla \cdot (a(x)\left|\nabla u\right|^{\alpha -1}\nabla u)'+c(x)\left|u\right|^{\alpha -1}u,\\ P_{\alpha }[v] & =\nabla \cdot (A(x)\left|\nabla v\right|^{\alpha -1}\nabla v)'+C(x)\left|v\right|^{\alpha-1}v,\end{aligned} \] where \(\alpha>0\) is a constant, and \(a,c,A,C\) are continuous (continuously differentiable) functions defined in a domain \(G\subset\mathbb{R}^n\). 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 \(\mathbb{R}^n\).

MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J70 Degenerate elliptic equations
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