ESAIM: PROCEEDINGS, October 2007, Vol.20, 53-62 Mohammed-Najib Benbourhim, Patrick Chenin, Abdelhak Hassouni & Jean-Baptiste Hiriart-Urruty, Editors EXISTENCE OF NONTRIVIAL SOLUTIONS FOR QUASI-LINEAR RESONANT PROBLEMS Abdesslem AYOUJIL 1 and Abdel Rachid El AMROUSS 1 Abstract. Combining the minimax arguments and the Morse Theory, by computing the critical groups at zero, we establish the existence of a nontrivial solution for a class of Dirichlet boundary value problems, with resonance at infinity and zero. R´ esum´ e. Par un proc´ ed´ e de minimax et application de la Th´ eorie de Morse, en calculant les groupes critiques en z´ ero, nous ´ etablissons l’existence d’une solution non triviale pour une classe de probl` emes de Dirichlet, avec r´ esonance `a l’infini et en z´ ero.. Introduction This paper is mainly concerned to study some classes of resonant elliptic equations. More specifically, we deal with the problem (P )  −Δ p u = f (x, u), in Ω, u =0, on ∂ Ω, where Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω and Δ p u := div(|∇u| p−2 ∇u), 1 <p< ∞, is the p-Laplacian. We assume that f :Ω × R → R is a Carath´ eodory function with subcritical growth, that is, (f 0 ) |f (x, t)|≤ c(1 + |t| q−1 ), ∀t ∈ R, a.ex ∈ Ω, for some c> 0, and 1 ≤ q<p ∗ where p ∗ = Np N−p if 1 <p<N and p ∗ =+∞ if N ≤ p. The growing attention in the study of the p-Laplace operator is widely motivated by the fact that it aries in various applications, e.g. non-Newtonian fluids, reaction-diffusion problems, flow through porus media, nonlinear elasticity, theory of superconductors, petroleum extraction, glacial sliding, astronomy, biology etc... The problem (P ) at resonance with p = 2 has been studied by few authors. Via directly variational methods or the minimax method, such that as the well-known saddle point theorem or the mountain pass theorem, solvability results for one solution were obtained, we refer to [6, 9, 11, 15] and references therein. Morse Theory, developed by Chang (cf. [7]) or Mawhin and willem (cf. [13]), is widely used in the study of the existence and multiplicity of solutions of certain nonlinear differential equations arising in the calculus of variations see for example [12]. Thus the computation of critical groups may yield the existence of nontrivial solution to the problem (P ). 1 Department of Mathematics, Faculty of Sciences, University Mohammed I, Oujda, Morocco ; e-mail: abayoujil@yahoo.fr & amrouss@sciences.univ-oujda.ac.ma c  EDP Sciences, SMAI 2007 Article published by EDP Sciences and available at http://www.edpsciences.org/proc or http://dx.doi.org/10.1051/proc:072005