Half-eigenvalues of elliptic operators

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6 Citations (Scopus)

Abstract

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).

Original languageEnglish
Pages (from-to)1439-1451
Number of pages13
JournalProceedings of the Royal Society of Edinburgh, Section A: Mathematics
Volume132
Issue number6
Publication statusPublished - 2002

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