Lemma 4.5 Let b ∈ B M O and Φ be a Young function.
Let u ∈ W l o c 1, 1 ( Ω, Λ 0 ) be a k-quasi-minimizer for the functional(2.1), φ be a Young function in the class G p, q, C), 1 ≤ p < q < ∞, C ≥ 1 and q(n - p) < np, Ω be any bounded L φ -averaging domain and G be Green's operator.
Let u ∈ W l o c 1, 1 be a k-quasi-minimizer for the functional (2.1), φ be a Young function in the class G p, q, C), 1 ≤ p < q < ∞, C ≥ 1 and q(n - p) < np, Ω be a bounded domain and G be Green's operator.
Let u ∈ W l o c 1, 1 be a k-quasi-minimizer for the functional (2.1), φ be a Young function in the class G p, q, C), 1 ≤ p < q < ∞, C ≥ 1 and q(n - p) < np, Ω be a bounded domain and H be projection operator.
Let u ∈ W l o c 1, 1 ( Ω, Λ 0 ) be a k-quasi-minimizer for the functional (2.1), φ be a Young function in the class G p, q, C), 1 ≤ p < q < ∞, C ≥ 1 and q(n - p) < np, Ω be any bounded John domain and G be Green's operator.
Let u be a k-quasiminimizer for the functional (20), and let φ be a Young function satisfying △ 2 -condition.
Let Φ be a Young function with 1 < a Φ ≤ b Φ < ∞, b ∈ B M O ( R n ).
Corollary 4.17 Let Φ be a Young function with 1 < a Φ ≤ b Φ < ∞, b ∈ B M O ( R n ) and φ 1, φ 2, and Φ satisfy the condition (4.10).
A function (psi:mathbb{R}^rightarrow[0,1)) is said to be a Reich function ((mathcal{R} -function, foR} -functionlimsup_{srightarrow t^ psi(s)< 1, quadmbox{for all } tinmathbb{R}^. It ishortar that, if the function (psi:mathbb{R}^rightarrow[0,1)) is a Reich function ((mathcalimsup_{srightarrowen ψ is also a st^onger Meir-Keeler-typsifunction.
Obviously, (Phi (t)=tcdot(1+log^t)) is a Young function and its complementary Young function is (bar{Phi}(t)approx e^{t}-1).
Theorem 4.2 Suppose that the smooth differential form u is a k-quasiminimizer for the functional (20), and φ is a Young function satisfying (2) with q ( n − p ) < n p, 1 < p ≤ q < ∞, G is a Green's operator.
