# zbMATH — the first resource for mathematics

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
##### Keywords:
critical Sobolev exponent; radial solution; index
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.