On the generalized spectrum for second-order elliptic systems

We consider the system of homogeneous Dirichlet boundary value problems (*) (FORMULA PRESENTED) in a smooth bounded domain W Í R^N, where L₁ and L₂ are formally selfadjoint second-order strongly uniformly elliptic operators. Using linear perturbation theory, continuation methods, and the Courant-Hilbert variational eigenvalue characterization, we give a detailed qualitative and quantitative description of the real generalized spectrum of (*), i.e., the set ((l, m) Î R²: (*) has a nontrivial solution). The generalized spectrum, a term introduced by Protter in 1979, is of considerable interest in the theory of linear partial differential equations and also in bifurcation theory, as it is the set of potential bifurcation points for associated semilinear systems.