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The index of the radial solution of some elliptic P.D.E. (English) Zbl 0717.35027
For \(\Omega_{\epsilon}=\{x\in {\mathbb{R}}^ N:\) \(1<| x| <1+\epsilon \}\), the following boundary value problem is \(considered:\)
P\({}_{\epsilon}:\) \(-\Delta u=u^{(N+2)/(N-2)}\) and \(u>0\) in \(\Omega_{\epsilon}\), \(u=0\) on \(\partial \Omega_{\epsilon}.\)
The corresponding energy functional is denoted by \(J_{\epsilon}(u)\). It is shown that problem \(P_{\epsilon}\) has one and only one radial solution \(u_{\epsilon}\). It is presented explicitly. For small \(\epsilon\), an approximation to \(J_{\epsilon}(u_{\epsilon})\) is given. Finally, a result concerning the index of \(u_{\epsilon}\) as critical point of \(J_{\epsilon}(u)\) is established.
Reviewer: W.Velte

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
47J25 Iterative procedures involving nonlinear operators
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