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Extrema problems with critical Sobolev exponents on unbounded domains. (English) Zbl 0851.49004

This paper is devoted to the minimization problem

𝒫 a :=minimize(u):= Ω |u| p +a|u| q ,ontheconstraintu𝒟 0 1,p (Ω), Ω |u| p * =1,

where 1<p<N, pq<p * :=pN/(N-p) and aL p * /(p * -q) (Ω). Here 𝒟 0 1,p (Ω) is the closure of 𝒟(Ω) with respect to the norm |u|=|u| p * +|u| p . The open set Ω can be unbounded. The main result of the paper states that under the assumption a0 and q>(N+1)p 2 -2Np (N-p)(p-1) if pN, then S a :=inf𝒫 a is achieved by a nonnegative function. They also prove that the inequality S a S 0 always holds. In particular, if a0 and a0, then S a =S 0 and S a is never achieved. Moreover, the inequality S a <S 0 holds under the assumptions of the above theorem.

49J20Optimal control problems with PDE (existence)