Morse index and uniqueness for positive solutions of radial <IMG BORDER="0" SRC="/tran/2004-356-11/S0002-9947-04-03628-1/gif-title0/img1.gif" >-Laplace equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Amandine Aftalion Filomena Pacella

Institution:	Laboratoire Jacques-Louis Lions, B.C. 187, Université Paris 6, 175 rue du Chevaleret, 75013 Paris, France ; Dipartimento di Matematica, Università di Roma ``La Sapienza", P.le A. Moro 2, 00185 Roma, Italy

Abstract:	We study the positive radial solutions of the Dirichlet problem $\Delta_p u+f(u)=0$ in , in , on $\partial B$ , where $\Delta_p u=\operatorname{div}(\vert\nabla u\vert^{p-2}\nabla u)$ , , is the -Laplace operator, is the unit ball in $\mathbb{R} ^n$ centered at the origin and is a 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 is of the type and $p\geq 2$ .

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