Summability of Solutions of the Dirichlet Problem for Nonlinear Elliptic Equations with Right-Hand Side in Classes Close to L~1

摘要

Suppose given n G N, n ≥ 2, a nonempty bounded open set Ω in R~n, and p e (1,n). Assume that c_1,c_2 > 0, g_1 ∈ L~(p/(p-1))(Ω), g_1 ≥ 0 in Ω, and a_i: Ω × R × R~n → R, i = 1,...,n, are Caratheodory functions. We also assume that, for almost all x ∈ Ω and any s G R and ξ ∈ R~n, the inequalities n Σ i=1 |a_i(x,s,ξ)| ≤ c_1|ξ|~(p-1)+ g_1(x), n Σ i=1 a_i(x,s,ξ)ξ_i ≥ c_2|ξ|~p hold and, for almost all x ∈ Ω and any s ∈ R and ξ, ξ′ ∈ R~n, ξ ≠ ξ′, the inequality n Σ i=1 [a_i(x,s,ξ)-a_i(x,s,ξ′)](ξ_i-ξ_i_′>0 holds.