Existence of a solution to a nonlinear equation

Equation (−Δ+k2)u+f (u) = 0 in D, u| ∂D = 0, where k = const > 0 and D ⊂ R3 is a bounded domain, has a solution if f :R → R is a continuous function in the region |u| a, piecewisecontinuous in the region |u| a, with finitely many discontinuity points uj such that f (uj ± 0) exist, and uf (y) 0 for |u| a, where a 0 is an arbitrary fixed number.  2005 Elsevier Inc. All rights reserved