This paper is concerned with the bifurcation from infinity of the nonlinear Schrödinger equation $-\Delta u+V\left(x\right)u=\lambda u+f(x,u),x\in {\mathbb{R}}^N.$ We treat this problem in the framework of dynamical systems by considering the corresponding parabolic equation on unbounded domains. Firstly, we establish a global invariant manifold for the parabolic equation on ${\mathbb{R}}^N$. Then, we restrict the parabolic equation to this invariant manifold, which generates a system of finite dimension. Finally, we use the Conley index theory and the shape theory of attractors to establish some new results on bifurcations from infinity and multiplicity of solutions of the Schrödinger equation under an appropriate Landesman–Lazer type condition.

本文关注的是非线性薛定谔方程从无穷大的分岔 $-\Delta 你+伏\left(X\right)你=\lambda 你+F(X,你),X\in {\mathbb{电阻}}^N.$我们通过考虑无界域上的相应抛物线方程，在动力系统的框架中处理这个问题。首先，我们为抛物线方程建立一个全局不变流形${\mathbb{电阻}}^N$. 然后，我们将抛物线方程限制为这个不变流形，它生成一个有限维的系统。最后，我们使用 Conley 指数理论和吸引子的形状理论，在适当的 Landesman-Lazer 类型条件下建立了关于薛定谔方程的无穷解和解的多重分岔的一些新结果。