# Article

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Keywords:
Morse theory in Banach spaces, p-laplace equations, critical exponent, critical groups, multiplicity, perturbation results, functionals with lack of smoothness, generalized Morse index
Summary:
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. Denoting by ${\mathcal P}_1(\O)$ the Poincar\´e polynomial of $0$, 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 $(P_n)_{n \in \N}$ distinct solutions. These results have been proved in [12] only as regards the case p 2, while they will be completely proved in the forthcoming work [13] in the case $p\geq2$, while they will be completely proved in the forthcoming work \cite{cvip} in the case $p\in (1,2)$. Note that, when $p\geq2$, the Euler functional associated to $(P_\l)$ is never $C^2$, so the approach already used for $p\geq 2$ fails. This problem will be faced exploiting recent results given in [7] and [8].
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