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Picone-type inequalities for half-linear elliptic equations and their applications. (English) Zbl 1089.35019
Picone-type inequalities for some degenerate quasilinear elliptic operators are established and Sturmian comparison theorems are derived as applications extending previous work by the authors [J. Inequal. Appl. 6, No. 4, 387–404 (2001; Zbl 1088.35024)]. The equations considered are of the form $$L[v]=f(x)$$ and $$\tilde L[v]=0$$, where $L[v]=\text{div } (A(x)|\nabla v|^{\alpha-1} \nabla v)+C(x)|v|^{\beta-1}v$ and $\tilde L[v]=\text{div }(A(x)|\nabla v|^{\alpha-1}\nabla v)+C(x)|v|^{\beta-1}v+D(x)|v|^{\gamma-1}v.$ The functions $$A(x)$$, $$C(x)$$, $$D(x)$$ are defined and positive in some bounded piecewise smooth domain $$G\subset \mathbb R^n$$, and $$0<\gamma<\alpha<\beta$$. Moreover, $$A\in C^1(\overline{G})$$, $$C,D,f\in C(\overline{G})$$. Oscillation results are also obtained by using Sturmian comparison theorems.

##### MSC:
 35J60 Nonlinear elliptic equations 35B45 A priori estimates in context of PDEs