, .Voniincan Ana&rrr. Tkheory. .Mcrkods & Apphnonc. Vol. 9. So 12. pp. 13X-1330. 1985 036?-546X. R5 33.110 T Ml Pnnted in Great Briram. 0 1985 Pergamon Press Ltd. REMARKS AND GENERALIZATIONS RELATED TO A RECENT MULTIPLICITY RESULT OF A. LAZER AND P. McKENNA BERNHARD RUF Forschungsinstitut fiir IMathematik, ETH-Zentrum, 8001 Ziirich, Switzerland zyxwvutsrqponmlkjihgfedc (Received 25 February 1981; received for publication 25 February 1985) Key words and phrcrres: Nonlinear Sturm-Liouville equation, multiple solutions, global bifurcation, nonlinear eigenvalues INTRODUCTION IN A RECENT paper Lazer and McKenna [6] considered ordinary differential equations of the following type _u” = g(u)--&?, +h, O<x<n u(0) = U(X) = 0, 1 (1) where g(s) = h* - p- + b(s), with S- = max{s, 0}, S- = max{-s, 0}, and b E C(R) satisfying lb(s)] s C(js1q + 1) with q < 1. Furthermore, el is the positive and normed eigenfunction corresponding to the first eigenvalue ,$ of Lu: = -u” with boundary conditions u(O) = u(,?r) = 0, and h E C(0, ;c> is given. They prove THEOREM 1. Assume Akc I > A > Akr and A1 > ,u > 0. Then (1) has at least 2k solutions provided that t is sufficiently large. We will show here that this result has a natural generalization to the case that ,u 5 Ai. To state the result we consider the model problem Lu = Au’ - Au- - tel, (2) where 0 < ,u < A, u+ = max{u, 0}, u- = u+ - u. We remark that the condition y C A is no restriction, cf. remark 6 below. By setting y = A - y, we can write (2) as Lu - yyu- = Au - rel. (3) If we now fix y > 0 and set t = 0 we have a nonlinear eigenvalue problem Lu - yu- = Au. (4) The following proposition is known (cf. FuEik [3], Dancer [2], and also [lo] where the result appears in the formulation given below). PROPOSITION 2. The values A for which (4) has a nontrivial solution form a sequence of pairs (Af, A$) E R’ X R’ which are ordered as follows O<~tr~,<;l:E~l+y<~i=~:<~:<A~ <A:=#<. . . <~.q,=~:,<n:,t,<~:,.,<. . .++ x, 1325