  We study the existence of solutions in Hilbert space $H$ of the semilinear
equation \[ L u+N(u)=h, \] where $L$ is linear self-adjoint, $N$ is a nonlinear
operator and $h\in H$. We concentrate on the case when $0$ is a right boundary
point of a gap in the spectrum of $L$ and an element of essential spectrum. The
sufficient conditions for solvability are based on monotonicity and sign
assumptions on operator $N$, and its behaviour on $\ker L$. We illustrate the
main theorem by an application to the study of nonlinear stationary
Schr\"{o}dinger equation on $\mathbb{R}^n$.
