Quasilinear elliptic equations involving measure valued absorption terms and measure data

Abstract

AbstractLet 1 < p < N and Ω ⊂ ℝN be an open bounded domain. We study the existence of solutions to equation $$(E) - {\Delta _p}u + g(u)\sigma = \mu $$ ( E ) − Δ p u + g ( u ) σ = μ in Ω, where g ∈ C(ℝ) is a nondecreasing function, μ is a bounded Radon measure on Ω and σ is a nonnegative Radon measure on ℝN. We show that if σ belongs to some Morrey space of signed measures, then we may investigate the existence of solutions to equation (E) in the framework of renormalized solutions. Furthermore, imposing a subcritical integral condition on g, we prove that equation (E) admits a renormalized solution for any bounded Radon measure μ. When $$g(t) = |t{|^{q - 1}}t$$ g ( t ) = | t | q − 1 t with q > p − 1, we give various sufficient conditions for the existence of renormalized solutions to (E). These sufficient conditions are expressed in terms of Bessel capacities.