By Froberg R.

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**Example text**

Next,we prove the invariance of the set H. For this aim we note that, by the embedding H01 → Lq , we have u q ≤ C ∇u , 2 (7) 2n for 2 ≤ q ≤ n−2 if n ≥ 3, q > 2 if n = 1,2 where C = C(n,q,Ω) is the best constant. 1. (Nakao[11]) Let ϕ(t) be a nonincreasing and nonnegative function deﬁned on [0, T ], T > 1, satisfying ϕ1+r (t) ≤ k0 (ϕ(t) − ϕ(t + 1)), t ∈ [0, T ] , for k0 > 1 and r ≥ 0. Then we have , for each t ∈ [0, T ], ϕ(t) ≤ ϕ(t) ≤ + ϕ(0)e−k[t−1] , r=0 ϕ(0)−r + k0 r [t − 1] + 0 where [t − 1]+ = max {t − 1, 0} and k = ln( k0k−1 ).

If u ∈ W 2,s for some s ≥ p , may we prove that u ∈ W 2,r for some r ≥ s? In [3] we prove the following result. 1. 10) and assume that D2 u ∈ Ls . 12) where r = φp (s) := 6s . (5 − p) s + 3 (p − 2) In particular, u ∈ W 2,s ⇒ u ∈ W 2,r . The above proposition allows us, by a bootstrap argument, to make any ﬁnite number of regularizing steps. , r is the ﬁxed point l = φp (l) of the map φp ). This requires sharp estimates at each stage of the proof. We succeed in proving these estimates and the desired result.

References [1] C. Bandle and R. Benguria, The Brezis-Nirenberg Problem on Sn , J. Diﬀ. Equ. 178 (2002), 264–279. [2] C. Bandle, S. Stingelin and Juncheng Wei, Multiple clustered layer solutions for semilinear elliptic problems on Sn , in preparation. [3] H. A. Peletier, Elliptic equations with critical exponent on S3 : new non-minimising solutions, C. R. A. S. , 339 (2004), 391–394. [4] H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm.