Existence of solutions for some quasilinear elliptic system with weight and a right-hand side given by the sum of a measure and a function

Abstract

We prove the existence of a solution u for the nonlinear elliptic system:

-divσ(x; u; Du)
u =
= µ + g(x; u) in Ω
0 on @Ω

where µ is Radon measure on Ω with finite mass and g satisfies some standards continuity
and growth conditions. In particular, we show that if the coercivity rate of σ lies in
the range ]s+1 s ; (s+1 s )(2 - n1 )] then u is approximately differentiable and the equation
holds with Du replaced by apDu. The proof relies on an approximation of µ by smooth
functions fk and a compactness result for the corresponding solutions uk. This follows
from a detailed analysis of the Young measure fδu(x) ⊗ #(x)g generated by the sequence
(uk; Duk), and the div-curl type inequality < #(x); σ(x; u; ·) >≤ σ(x) < #(x); : > for the
weak limit σ of the sequence.