Reduced limits for nonlinear equations with measures

Abstract

Listen

Translate article

We consider equations ( E) − Δ u + g ( u ) = μ in smooth bounded domains Ω ⊂ R N , where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures ( μ k ) , assume that for each k ⩾ 1 there exists a solution u k of ( E) with datum μ k and zero boundary data. We show that if u k → u # in L 1 ( Ω ) , then u # is a solution of ( E) relative to some finite measure μ # . We call μ # the reduced limit of ( μ k ) . This reduced limit has the remarkable property that it does not depend on the boundary data, but only on ( μ k ) and on g. For power nonlinearities g ( t ) = | t | q − 1 t , ∀ t ∈ R , we show that if ( μ k ) is nonnegative and bounded in W − 2 , q ( Ω ) , then μ and μ # are absolutely continuous with respect to each other; we then produce an example where μ # ≠ μ .