Solutions to sublinear elliptic equations with finite generalized energy

Abstract

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation Lu=σuq+μinΩ,in the sublinear case 0 < q< 1 , with finite generalized energy: Eγ[u] : = ∫ Ω| ∇ u| 2uγ-1dx< ∞, for γ> 0. In this case u∈ Lγ+q(Ω , σ) ∩ Lγ(Ω , μ) , where γ= 1 corresponds to finite energy solutions. Here Lu:=-div(A∇u) is a linear uniformly elliptic operator with bounded measurable coefficients, and σ, μ are nonnegative functions (or Radon measures), on an arbitrary domain Ω ⊆ Rn which possesses a positive Green function associated with L. When 0 < γ≤ 1 , this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space W˙01,p(Ω) for 1 < p≤ 2.