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

Abstract:

We study the positive radial solutions of the Dirichlet problem $\Delta _p u+f(u)=0$ in $B$, $u>0$ in $B$, $u=0$ on $\partial B$, where $\Delta _p u=\operatorname {div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, is the $p$-Laplace operator, $B$ is the unit ball in $\mathbb {R}^n$ centered at the origin and $f$ is a $C^1$ function. We are able to get results on the spectrum of the linearized operator in a suitable weighted space of radial functions and derive from this information on the Morse index. In particular, we show that positive radial solutions of Mountain Pass type have Morse index 1 in the subspace of radial functions of $W_0^{1,p}(B)$. We use this to prove uniqueness and nondegeneracy of positive radial solutions when $f$ is of the type $u^s+u^q$ and $p\geq 2$.