On the simplicity and uniqueness of positive eigenvalues admitting positive eigenfunctions for weakly coupled elliptic systems (Q1124040)

The boundary value problem for the weakly coupled linear system \[ (1)\quad Lu=PMu\quad in\quad \Omega,\quad u\equiv 0\quad on\quad \delta \Omega \] is studied. L is a diagonal matrix with elements \(L^ i\) \((i=1,...,r)\) which are formally selfadjoint uniformly elliptic differential operators of second order. The off-diagonal elements of the matrix M are non-negative functions. P is a non-negative r*r scalar matrix with \(p_{ii}>0\) for \(i=1,...,r\); special case and, of course, main object of consideration is \(P:=\lambda I\), \(\lambda >0.\) The author is mainly interested in choices of P which admit classical solutions \(u:=(u^ 1,...,u^ r)\) of (1) for which \(u^ i(x)\geq 0\) on \({\bar \Omega}\), \(i=1,...,r\). He shows that simplicity and uniqueness results obtained in a previous work [see the author and \textit{K. Schmitt}, SIAM J. Math. Anal. 17, 850-862 (1986; Zbl 0606.35033)] can be proved without the commutativity assumption \(L^{-1}M=ML^{-1}\) provided that L is formally selfadjoint and M is symmetric. Thus results of \textit{A. Manes} and \textit{A. M. Micheletti} [Boll. Unione Mat. Ital., IV. Ser. 7, 285-301 (1973; Zbl 0275.49042)] are extended to systems.