Multiplicity results for a semilinear elliptic problem with crossing of multiple eigenvalues
Celius A. Magalhães · Differential and Integral Equations · 1991
Let 0 C ~R_N be a smooth bounded domain and consider the Dirichlet problem -!:iu=f(u)+h inn, u=Oonon (0.1) where hE £ 2 (0) is a given function and f E C(JR.) satisfies lim f(s)/s=(3 S---+-00 lim f(s)/s=a.s-+oo Let .A1 0 is small enough, then (a, (3) = (.Aj + E, Aj -t:) belongs to the so-called Ambrosetti-Prodi region.In this case, with additional hypotheses on f and for appropriate h E £ 2 (0), she obtained the existence of at least two solutions of problem (0.1).The authors in [1], with slightly stronger hypotheses, showed that problem (0.1) has exactly two solutions.The authors of [6] and [10] extended the result in [9], and [11] extended the result in [1].In this paper we consider also the case when (a, (3) = ( Aj + E, Aj -E), but allowing Aj to have multiplicity m > 1.In the case that m is even and a condition on the eigenspace is satisfied, we obtain the existence of four solutions.The author is grateful to D.G. de Figueiredo for several discussions during this work, and also to the referee for the example given at the end of the paper.1.The main result.Let Ak and ¢Jk, k = 1, 2, ... , be as in the Introduction, and assume that Aj = Aj+l = • • • = Aj+m-l is an eigenvalue of multiplicity m.Given a direction (a, b) E JR. 2 we will study problem (0.1) for (a, (3) = (.Aj + w, Aj + t:b) and E > 0 sufficiently small.