Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data

Abstract

Let un be the sequence of solutions of -div(a(x, un, ∇un)) + unq-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 Lloc1 (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. © 2002 Elsevier Science (USA).