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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.

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