Nonlinear Elliptic Equations in RN without Growth Restrictions on the Data

We show existence and regularity of solutions in RN to nonlinear elliptic equations of the form −div A(x, Du) + g(x, u) = ƒ when ƒ is just a locally integrable function, under appropriate growth conditions on A and g but not on ƒ. Roughly speaking, in the model case −Δp(u) + us−1u = ƒ with p > 2 − (1/N), existence of a nonnegative solution in RN is guaranteed for every nonnegative ƒ L1loc(RN) if and only if s > p − 1.