Let Ω ⊂ ℝ n , for n ≥ 2 , be a bounded C 2 domain. Let q ∈ L l o c 1 ( Ω ) with q ≥ 0 . We give necessary conditions and matching sufficient conditions, which differ only in the constants involved, for the existence of very weak solutions to the boundary value problem ( - ▵ - q ) u = 0 , u ≥ 0 on Ω , u = 1 on ∂ Ω , and the related nonlinear problem with quadratic growth in the gradient, - ▵ u = | ∇ u | 2 + q on Ω , u = 0 on ∂ Ω . We also obtain precise pointwise estimates of solutions up to the boundary. A crucial role is played by a new “boundary condition” on q which is expressed in terms of the exponential integrability on ∂ Ω of the balayage of the measure δ q d x , where δ ( x ) = dist ( x , ∂ Ω ) . This condition is sharp, and appears in such a context for the first time. It holds, for example, if δ q d x is a Carleson measure in Ω , or if its balayage is in B M O ( ∂ Ω ) , with sufficiently small norm. This solves an open problem posed in the literature.