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 is a continuous function in the region |u|⩾a, piecewise-continuous 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.