Advances in Differential Equations Volume 1, Number 6, November 1996, pp. 1025 – 1052 SIGN CHANGING SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS Monica Musso and Donato Passaseo Dipartimento di Matematica, Universit` a di Pisa, Via Buonarroti, 2, 56127 Pisa, Italy (Submitted by: E.N. Dancer) Abstract. This paper is concerned with a class of nonlinear elliptic Dirichlet problems ap- proximating degenerate equations. If the degeneration set consists of k connected components, by using variational methods, it is proved the existence of k 2 distinct nodal solutions, having exactly two nodal regions, whose positive and negative parts concentrate near subsets of the degeneration set. 1. Introduction. In this paper we are concerned with the existence and multiplicity of solutions for the problem P " 8 > < > : div (a " (x)Du)+ g(x, u)=0 in Ω u + 6⌘ 0, u - 6⌘ 0 in Ω u =0 on @ Ω, where Ω is a smooth bounded domain of R N with N ≥ 1, for all " > 0, a " (x)=(a i,j " (x)) is a positive defined symmetric N ⇥ N matrix with coefficients a i,j " in L 1 (Ω; R), and g : Ω ⇥ R ! R is a given function; u + and u - denote respectively the positive and the negative part of u. We require that the matrix a " (x) degenerates as " ! 0 in some subsets Ω 1 ,... , Ω k of Ω; g(x, t) is required to be a superlinear function with subcritical growth (the conditions on a " (x), g(x, t) and the degeneration subsets Ω 1 ,... , Ω k are precised in Section 2). We refer the reader to the introduction of [9] for a more detailed discussion on the connection of this problem with many other problems, which have been investigated very much in recent years, where the multiplicity of the solutions is related to some concentration phenomena. Here we recall only that for elliptic problems approximating degenerate equations, like P " , these concentration phenomena were first pointed out in [11]; in that paper it is proved that, as " ! 0, every solution of problem P " tends to “concentrate itself” near the degeneration set of a " (x). This property allows us to obtain in [9] a multiplicity result of positive solutions, which is related to the geometrical properties of the degeneration set: if, for example, it consists of several connected components Ω 1 ,... , Ω k such that Ω \ k S t=1 Ω t is a connected domain, then problem P " , for " > 0 small enough, has several positive solutions that concentrate themselves, as " ! 0, near different connected components Ω 1 ,... Ω k of the degeneration set. Received for publication February 1996. AMS Subject Classifications: 35J65, 35J20, 35J70. 1025