带零点的半线性方程的可解性及其在非线性薛定谔方程中的应用
Solvability of semilinear equations with zero on the boundary of spectral gap and applications to nonlinear Schrödinger equation
摘要：
本文研究了半线性方程\[ L u+N(u)=h, \]在Hilbert空间$H$中解的存在性，其中$L$是线性自伴的，$N$是非线性算子，$h\in H$ .我们集中在$0$是$L$的谱中的间隙的右边界点和本质谱的元素的情况下。可解性的充分条件是基于算子$N$的单调性和符号假设，以及它在$\ker L$上的行为。通过在$\mathbb{R}^n$上对非线性定常薛定谔方程的研究来说明主要定理.
英文摘要：
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 Schrödinger equation on $\mathbb{R}^n$.
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