Sturmian theorems for second order systems

Abstract:

Sturmian theorem are established for weakly coupled elliptic systems generated in a bounded domain by the expressions ${l_1}\vec u = - \Delta \vec u + A\vec u,{l_2}\vec w = - \Delta \vec w + B\vec w$, and Dirichlet boundary conditions. Here $\Delta$ denotes the Laplace operator, and $A,B$ are $m \times m$ matrices. We do not assume that $A,B$ are symmetric, but instead essentially require $B$ irreducible and ${b_{ij}} \leqslant 0{\text { if }}i \ne j$. Estimates on the real eigenvalue of ${l_2}$, with a positive eigenvector are then obtained as applications. Our results are motivated by recent theorems for ordinary differential equations established by Ahmad, Lazer and Dannan.