Dirichlet Boundary Value Problem for the Second Order Asymptotically Linear System

Abstract

We consider the second order system ${\mathbf{x}}^{\mathrm{\prime }\mathrm{\prime }}=\mathbf{f}(\mathbf{x})$ with the Dirichlet boundary conditions $\mathbf{x}(\mathrm{0})=\mathbf{0}=\mathbf{x}(\mathrm{1})$, where the vector field $\mathbf{f}\in C^{\mathrm{1}}({\mathbb{R}}^n,{\mathbb{R}}^n)$ is asymptotically linear and $\mathbf{f}(\mathbf{0})=\mathbf{0}$. We provide the existence and multiplicity results using the vector field rotation theory.