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 $\mathbb{ R}^n$.
Categories: math.FA math.AP