Abstract. We obtain necessary and sufficient conditions for the existence of a positive finite energy solution to the inhomogeneous quasilinear elliptic equationin the sub-natural growth case 0 < q < p − 1, where ∆ p (1 < p < ∞) is the p-Laplacian, and σ, µ are positive Borel measures on R n . Uniqueness of such a solution is established as well.Similar inhomogeneous problems in the sublinear case 0 < q < 1 are treated for the fractional Laplace operator (−∆) α in place of −∆ p , on R n for 0 < α < n 2 , and on an arbitrary domain Ω ⊂ R n with positive Green's function in the classical case α = 1.

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation Lu = σu q + µ in Ω, in the sublinear case 0 < q < 1, with finite generalized energy: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 ⊆ R n 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 spaceẆ 1,p 0 (Ω) for 1 < p ≤ 2.2010 Mathematics Subject Classification. Primary 35J61, 42B37; Secondary 31B10, 31B15.

We study the existence of positive solutions to quasilinear elliptic equations of the typein the sub-natural growth case 0 < q < p − 1, where ∆ p u = ∇ · (|∇u| p−2 ∇u) is the p-Laplacian with 1 < p < n, and σ and µ are nonnegative Radon measures on R n . We construct minimal generalized solutions under certain generalized energy conditions on σ and µ. To prove this, we give new estimates for interaction between measures. We also construct solutions to equations with several sub-natural growth terms using the same methods.✩ This is a pre-print of an article published in Nonlinear Analysis. The final authenticated version is available online at: https://doi.

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We study the existence problem for positive solutions u ∈ L r (R n), 0 < r < ∞, to the quasilinear elliptic equation −∆ p u = σu q in R n in the sub-natural growth case 0 < q < p − 1, where ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian with 1 < p < ∞, and σ is a nonnegative measurable function (or measure) on R n. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators in place of ∆ p such as the A-Laplacian divA(x, ∇u), or the fractional Laplacian (−∆) α on R n , as well as linear uniformly elliptic operators with bounded measurable coefficients div(A∇u) on an arbitrary domain Ω ⊆ R n with a positive Green function.