Nonlinear Elliptic Equations in RN without Growth Restrictions on the Data

Abstract

We show existence and regularity of solutions in R N 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 −Δ p ( u) + | u| s−1 u = ƒ with p > 2 − (1/ N), existence of a nonnegative solution in R N is guaranteed for every nonnegative ƒ ∈ L 1 loc( R N ) if and only if s > p − 1.