Integrability for Solutions to Some Nonhomogeneous Quasilinear Elliptic Problems

In this paper we prove an estimate for the measure of superlevel sets for weak solutions u of nonhomogeneous quasilinear elliptic systems − Xn i=1 Di i£« i£­ Xn j=1 X N β=1 a αβ ij (x, u(x))Dju β (x) i£¶ i£¸ = − Xn i=1 Dif α i (x, u(x)), (∗) α = 1, 2, · · · , N. The diagonal coefficients a γγ ij (x, y) are elliptic for large values of u, the off-diagonal coefficients are small when |u| is large, the faster offdiagonal coefficients decay, the higher integrability of u becomes.