Strong Stability Results for Solutions of Elliptic Equations with Power-like Lower Order Terms and Measure Data

Let un be the sequence of solutions of−div(a(x, un, ∇un))+|un|q−1 un=fn in Ω,un=0 on ∂Ω, where Ω is a bounded set in RN and fn is a sequence of functions which is strongly convergent to a function f in L1loc(Ω\K), with K a compact in Ω of zero r-capacity; no assumptions are made on the sequence fn on the set K. We prove that if a has growth of order p−1 with respect to ∇u(p>1), and if q>r(p−1)/(r−p), then un converges to u, the solution of the same problem with datum f, thus extending to the nonlinear case a well-known result by H. Brezis.