We consider general selfadjoint elliptic eigenvalue problems (P) Au = Xr(x)u, in an open set O C R*1. Here, the operator A is positive and of order 2m and the "weight" r is a function which changes sign in O and is allowed to be discontinuous. A scalar A is said to be an eigenvalue of (P) if Au = Xru-in the variational sense-for some nonzero u satisfying the appropriate growth and boundary conditions. We determine the asymptotic behavior of the eigenvalues of (P), under suitable assumptions. In

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... the case when Ü is bounded, we assumed Dirichlet or Neumann boundary conditions. When Q is unbounded, we work with operators of "Schrodinger type"; if we set r± = max(±r,0), two cases appear naturally: First, if f! is of "weighted finite measure" (i.e., fc¡(r+)k/2rn < +oo or fn(r-)k/2m < +oo), we obtain an extension of the well-known Weyl asymptotic formula; secondly, if f! is of "weighted infinite measure" (i.e., /n(r+)'c/2m = +co or /n(r_)fc/2m = +oo), our results extend the de Wet-Mandl formula (which is classical for Schrodinger operators with weight r = 1). When 0 is bounded, we also give lower bounds for the eigenvalues of the Dirichlet problem for the Laplacian. Fetnassi [F1F 1, F1F 2]. Key words and phrases. Elliptic boundary value problems, indefinite weight function, asymptotic behavior of eigenvalues, lower bounds of eigenvalues, spectral theory, variational methods, operators of Schrodinger type.