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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

l α [y]=(r(t)y ' α-1 y ' ) ' +q(t)y α-1 y,L α [y]=(R(t)z ' α-1 z ' ) ' +Q(t)z α-1 z,

where α>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

p[u]=·(a(x)u)+c(x)u,P[v]=·(A(x)v)+C(x)v,

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

p α [u]=·(a(x)u α-1 u) ' +c(x)u α-1 u,P α [v]=·(A(x)v α-1 v) ' +C(x)v α-1 v,

where α>0 is a constant, and a,c,A,C are continuous (continuously differentiable) functions defined in a domain G 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 α [u]=0 or P α [v]=0 in an unbounded domain in n .


MSC:
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35J70Degenerate elliptic equations