Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem Dimitri Mugnai Dipartimento di Matematica “G. Castelnuovo” Universit` a di Roma “La Sapienza” P.le Aldo Moro 2, 00185 Roma, Italy e-mail: mugnai@mat.uniroma1.it Abstract We consider a general nonlinear elliptic problem of the second order whose associated functional presents two linking structures and we prove the existence of three nontrivial solutions to the problem. 2000AMS subject classification: 35J65, 35J20, 49J40 Keywords and phrases: ∇–condition, linking, superlinear and subcritical equa- tions. 1 Introduction In this paper we consider the following problem (P )  -Δu - λu = g(x, u) in Ω, u =0 on ∂ Ω, where Ω is a smooth bounded domain of R N , N ≥ 3, λ ∈ R and g :Ω × R -→ R. Multiplicity results for solutions of nonlinear boundary value problems have been faced by a large number of authors, in different situations: sublinear, asymptot- ically linear, superlinear, subcritical, critical, supercritical... We are interested in a superlinear and subcritical problem (see Section 2). We will make the standard superlinear and subcritical assumptions on g ([4], [14], [22]) and we will show that for some values of λ, problem (P ) has at least three nontrivial solutions (the triv- ial solution being a solution of (P ) as well). Such a result seems to be new and, under quite general assumptions, it improves some previous results for analogous problems and it also parallels many results for different equations (see below). 1