Abstract
We prove a characterization of a nonhomogeneous A-harmonic equation and describe its generalization. We also point out its connection with 1-Harmonic equation.
Highlights
Both A-harmonic equations and p-harmonic geometry are rich subjects 1–5
0, 1.1 where |∇f| is the length of the gradient ∇f of f, and for a C2 function f without a critical point, div ∇f/|∇f| is said to be the 1-tension field of f
Journal of Inequalities and Applications where A : Ω x Rn → Rn satisfies i |A x, u | ≤ λw x |u|p−1, ii A x, u, u ≥ λ−1w x |u|p, where 1 < p < ∞, λ ≥ 1 are two fixed constants, and w x is called a weight if w ∈ L1loc Rn and w > 0 a.e
Summary
Both A-harmonic equations and p-harmonic geometry are rich subjects 1–5. Many results on both topics have been derived, respectively, but there are very few papers relating both subjects. We consider the following setting: a C1 function f : Rn → R is said to be A-harmonic if it is a weak solution of A-harmonic equation div A x, ∇f ∇f. Consider the following second-order divergencetype elliptic equation: div A x, ∇f x 0, Journal of Inequalities and Applications where A : Ω x Rn → Rn satisfies i |A x, u | ≤ λw x |u|p−1, ii A x, u , u ≥ λ−1w x |u|p, where 1 < p < ∞, λ ≥ 1 are two fixed constants, and w x is called a weight if w ∈ L1loc Rn and w > 0 a.e. in general dμ wdx where w is a weight.
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