Multiple positive solutions for a p-Laplace critical problem (p >1), via Morse theory.

We consider the quasilinear elliptic problem \[ (P_\lambda) \ \ \left\{ \begin{array}{ll} - \Delta _{p}u = \lambda u^{q-1} + u^{p^*-1} & \hbox{in} \ \Omega \\ u>0 & \hbox{in} \ \Omega \\ u=0 & \hbox{on} \ \partial \Omega \end{array} \right. \] where $\Omega$ is bounded in $\R^N\!$, $N \geq p^2, \, 1< p \leq q < p^*, \, p^*= \frac{Np}{N-p}%Np/(N-p) , \, \lambda >0$ is a parameter. \noindent Denoting by ${\mathcal P}_1(\O)$ the Poincar\'e polynomial of $\O$, we state that, for any $p>1$, there exists $\lambda^*>0$ such that, for any $\lambda\in (0,\lambda^*)$, either $(P_\lambda)$ has at least ${\mathcal P}_1(\O)$ distinct solutions or, if not, $(P_\l)$ can be approached by a sequence of problems $(P_n)_{n \in \N}$, each having at least ${\mathcal P}_1(\O)$ distinct solutions. These results have been proved in \cite{cvcr} only as regards the case $p\geq2$, while they will be completely proved in the forthcoming work \cite{cvip} in the case $p\in (1,2)$. \newline Note that, when $p\in (1,2)$, the Euler functional associated to $(P_\l)$ is never $C^2$, %as $ u\in W^{1,p}_0(\Omega) \, \mapsto \, \int_{\Omega} |\nabla u|^p \, dx \ $ is not $C^2$, so the approach already used for $p\geq 2$ fails. This problem will be faced exploiting %, via suitable approximations of $(P_\lambda)$, recent results given in \cite{cdvlin} and \cite{cdv}.