Morse index and uniqueness for positive radial solutions of $p$-Laplace equations


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Let us consider the Dirichlet problem $\D_p u+f(u)=0$ in $ B$, $u>0$ in $B$, $u=0$ on $ \partial B$, where $\D_p u=÷(|\nabla u|^{p-2}\nabla u)$, $p>1$ is the $p$-Laplace operator, $B$ is the unit ball in $\R^n$ centered at the origin and $f$ is a $C^1$ function. Introducing some suitable weighted space, we are able to get results on the spectrum of the linearized operator and derive information about the Morse index of radial solutions. Moreover we compute the critical groups and show that Mountain Pass solutions have Morse index 1. We use this to prove uniqueness of positive radial solutions when $f$ is of the type $u^s+u^q$ and $p\geq 2$.

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Date: 2002-09-30