Half-eigenvalues of elliptic operators

Suppose that L is a second-order self-adjoint elliptic partial differential operator on a bounded domain O ? Rⁿ, n = 2, and a, b ? L⁸(O). If the equation Lu = au⁺ - bu^- + ?u (where ? ? R and u^±(x) = max{±u(x),0}) has a non-trivial solution u, then ? is said to be a half-eigenvalue of (L; a, b). In this paper, we obtain some general properties of the half-eigenvalues of (L; a, b) and also show that, generically, the half-eigenvalues are 'Simple'. We also consider the semilinear problem Lu = f(x, u), where f: O × R ? R is a Carathéodory function such that, for a.e. x ? O, a(x) = lim _??8 f(x,?)/?, b(x) = lim _??-8 f(x,?)/? and we relate the solvability properties of this problem to the location of the half-eigenvalues of (L; a, b).