A Discrete Resonance Problem with Periodic Nonlinear Forcing

Abstract

In this paper, we study the problem $Ax=\lambda_1 x-cv(x), x\in\R^n$ where $A$ is a Laplacian matrix, $\lambda_1$ is the principal eigenvalue of $A$, $c\in\R$, $v(x)$ is a periodic gradient vector field, and $n=2$ or $3$. We show that the problem has infinitely many solutions with all positive components and infinitely many solutions with all negative components. With certain restrictions on $c$ we provide more detailed information about the solution set. In particular, for the $n=3$ case we prove that the solutions can be characterized as local minima and saddle points of the associated functional.