Abstract
A generalized nonlinear Picone identity for the p-biharmonic operator is established in this paper. As applications, a Sturmian comparison principle to the p-biharmonic equation with singular term, a Liouville’s theorem to the p-biharmonic system, and a generalized Hardy–Rellich type inequality are obtained.
Highlights
Introduction and resultsIn 1971, Dunninger [1] established a Picone identity div u∇(a u) – a u∇u – u2 ∇(A v) + A v∇ u2 v v u2 =–(A v) + u + (A – a)( u)2 v u u– 22 v +A ∇u – u∇v (1.1)v v v where u, v, a u, A v are twice continuously differentiable functions with v = 0 and a and A are positive weights
The purpose of this paper is to present a generalized nonlinear Picone identity for the p-biharmonic operator, which extends the results of Dwivedi and Tyagi [3] and Dwivedi [2]
We show a Liouville’s theorem for the p-biharmonic system by Theorem 1.5 as follows
Summary
With some simplifications in (1.1), recently, Dwivedi and Tyagi [3] have obtained the following linear Picone identity (see Theorem 1.1) for the biharmonic operator 2u = ( u) and gave several remarks on the qualitative questions such as Morse index and Hardy– Rellich type inequality. It is noteworthy that Dwivedi and Tyagi [4] established a Caccioppoli-type inequality by an application of Theorem 1.1. Dwivedi and Tyagi [5] extended the result of Theorem 1.1 on Heisenberg group and obtained its applications.
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