Existence of a solution to a nonlinear equation
  Equation $(-\Delta+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region $|u|\leq a$, with finitely many discontinuity points $u_j$ such that $f(u_j\pm 0)$ exist, and   $uf(y)\geq 0$ for $|u|\geq a$, where $a\geq 0$ is an arbitrary fixed number. 
A.G.Ramm