Multiplicity and nondegeneracy of positive solutions to quasilinear equations on compact Riemannian manifolds

We consider a compact, connected, orientable, boundaryless Riemannian manifold (M, g) of class C∞ where g denotes the metric tensor. Let n = dim M ≥ 3. Using Morse techniques, we prove the existence of $2{mathcal P}-1(M) -1$ nonconstant solutions u H1,p(M) to the quasilinear problem $$(P-epsilon) left{{egin{array}{@{}l@{}} -epsilon^p Delta-{p,g} u +u^{p-1}=u^{q-1}, \u>0,end{array}} ight$$ for ε > 0 small enough, where 2 ≤ p < n, p < q < p∗, p∗= np/(n - p) and $Delta-{p, g} u = extrm{div}-g (ert abla uert-{g}^{p-2} abla u)$ is the p-laplacian associated to g of u (note that Δ2,g = Δg) and ${mathcal P}-t(M)$ denotes the Poincaré polynomial of M. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem (Pε).