Eigenvalue problems for a cooperative system with a large parameter

Abstract

We consider the principal eigenvalue of a cooperative system of elliptic boundary value problems as a parameter tends to infinity of the form $$\begin{array}{rlrl}{\mathcal{A}}_1{u}_1+\lambda m_1{u}_1-d_1{u}_2& =\mu (\lambda )u_1& & \text{in }\mathrm{\Omega }\text{,}\\ {\mathcal{A}}_2{u}_2+\lambda m_2{u}_2-d_2{u}_1& =\mu (\lambda )u_2& & \text{in }\mathrm{\Omega }\text{,}\\ u_1=u_2& =0& & \text{on }\partial \mathrm{\Omega }\text{,}\end{array}$$ on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$ with ${\mathcal{A}}_1$, ${\mathcal{A}}_2$ uniformly strongly elliptic operators in divergence form and $d_1$, $d_2$ positive and $m_1$, $m_2$ non-negative and zero on some large enough set. We look at the limit problem as $\lambda \to \mathrm{\infty }$.

The main aim is to introduce an alternative approach to deal with the limit problem by focusing on the resolvent operator corresponding to the system rather than the eigenvalue problem itself. This allows the consistent use of elementary properties of bilinear forms and the semi-groups they induce. At the same time we weaken assumptions in related work by Álvarez Caudevilla & López-Gómez (2008) and Dancer (2011).