The index of the radial solution of some elliptic P.D.E. (English) Zbl 0717.35027

Bénichou, A.; Pomet, J.-B.

Nonlinear Anal., Theory Methods Appl. 14, No. 11, 991-997 (1990).

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

Keywords:

critical Sobolev exponent; radial solution; index

References:

[1]	Fowler, R.H., Further studies on Edmen’s and similar differential equations, Q. J. math., 2, 259-288, (1931) · Zbl 0003.23502
[2]	Berger, M.; Gauduchon, P.; Mazet, E., Spectre d’une variété riemannienne, () · Zbl 0223.53034
[3]	Brezis, H.; Nirenberg, L., Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Communs pure appl. math., 36, 437-477, (1983) · Zbl 0541.35029