Solutions to sublinear elliptic equations with finite generalized energy

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation Lu = suq + mu in , in the sublinear case 0 < q < 1, with finite generalized energy: E. [u] := |. u| 2u.- 1dx < 8, for. > 0. In this case u. L.+ q(, s) n L. (, mu), where. = 1 corresponds to finite energy solutions. Here Lu := - div(A. u) is a linear uniformly elliptic operator with bounded measurable coefficients, and s, mu are nonnegative functions (or Radon measures), on an arbitrary domain . Rn which possesses a positive Green function associated with L. When 0 <. = 1, this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space. W 1, p 0 () for 1 < p <= 2.