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

Solvability of semilinear equations with zero on the boundary of spectral gap and applications to nonlinear Schr\"{o}dinger equation