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

Authors: Amandine Aftalion and Filomena Pacella
Journal: Trans. Amer. Math. Soc. 356 (2004), 4255-4272
MSC (2000): Primary 58E05, 35J05
Published electronically: June 2, 2004
MathSciNet review: 2067118
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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 $\Delta_p u=\operatorname{div}(\vert\nabla u\vert^{p-2}\nabla u)$, $p>1$, 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 $p\geq 2$.

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Additional Information

Amandine Aftalion
Affiliation: Laboratoire Jacques-Louis Lions, B.C. 187, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France

Filomena Pacella
Affiliation: Dipartimento di Matematica, Università di Roma “La Sapienza", P.le A. Moro 2, 00185 Roma, Italy

Received by editor(s): May 23, 2002
Published electronically: June 2, 2004
Additional Notes: Research of the second author was supported by MIUR, project “Variational methods and Nonlinear Differential Equations”
Article copyright: © Copyright 2004 American Mathematical Society

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