We consider a compact, connected, orientable, boundaryless Riemannian manifold $(M,g)$ of class $C^infty$ where $g$ denotes the metric tensor. Let $n= dim M geq 3$. Using Morse techniques, we prove the existence of $2{mathcal P}_1(M) -1$ non-costant solutions $uin H^{1,p}(M)$ to the quasilinear problem [ (P_epsilon) left{ begin{array}{l} -epsilon^p , Delta_{p,g} u +u^{p-1}=u^{q-1} \ u>0 end{array} ight. label{eqab} ] for $varepsilon>0$ small enough, where $2 leq p<n$, $p < q <p^*$, $p^* = np/(n-p)$ and $Delta_{p,g} u = extrm{div}_g (|nabla u|_g^{p-2}nabla u)$ is the $p$-laplacian associated to $g$ of $u$ (note that $Delta_{2,g} = Delta_g$) and ${mathcal P}_t(M)$ denotes the Poincar'e Polynomial of $M$. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem $(P_varepsilon)$.

Regularity results and critical group estimates are studied for critical (p,r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.

This work deals with Morse index estimates for a solution $ uin H_1^p(M)$ of the quasilinear elliptic equation $ - extrm{div}_g left( left(alpha +|nabla u|_g^2 ight)^{(p-2)/2}nabla u ight)=h(x,u) $, where (M, g) is a compact, Riemannian manifold, 0 < alpha, 2 leq p < n. The nonlinear right-hand side h(x,s) is allowed to be subcritical or critical.

In this paper we illustrate the lineguides of our research group. We describe some recent results concerning the study of some nonlinear differential equations and systems having a variational nature and arising from physics, geometry and applied sciences. In particular we report existence, multiplicity and regularity results for the solutions of these nonlinear problems. We point out that, in treating the above problems, the used methods for finding solutions are variational and topological, indeed the existence of solutions of the considered equations is obtained searching for critical points of suitable functionals defined on manifolds embedded into infinite dimensional functional spaces, while the regularity of the solutions is studied by means of geometric and harmonic analysis tools.