Extremum Principle For Very Weak Solutions Of A-Harmonic Equation

This paper deals with the extremum principle for very weak solutions of A -harmonic equation, where the operator A satisfies the monotonicity inequality, the Lipschitz type condition and the homogeneity condition. The extremum principle for very weak solutions of the A -harmonic equation is derived by using the stability result such that if u(x) ∈ W ^1,r(Ω) be very weak solutions of the A-harmonic equation and m ≤ u(x) ≤M in the Sobolev sense, then m ≤ u(x) ≤ M almost everywhere in Ω,provided that r > r₁. As a corollary, the uniqueness result for the very weak solution of the 0-Dirichlet boundary value problem is also proved.