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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 $${\cal P}_a : = \cases \text{minimize } {\cal E} (u) : = \int_\Omega |\nabla u |^p + a |u |^q, \\ \text{on the constraint } u \in {\cal D}_0^{1,p} (\Omega),\ \int_\Omega |u |^{p^*} = 1, \endcases$$ where $1 < p < N$, $p \le q < p^* : = pN/(N - p)$ and $a \in L^{p^*/(p^* - q)} (\Omega)$. Here ${\cal D}_0^{1,p} (\Omega)$ is the closure of ${\cal D} (\Omega)$ with respect to the norm $|u |= |u |_{p^*} + |\nabla u |_p$. The open set $\Omega$ can be unbounded. The main result of the paper states that under the assumption $a \ngeq 0$ and $q > {(N + 1) p^2 - 2Np \over (N - p) (p - 1)}$ if $p \ge \sqrt N$, then $S_a : = \inf {\cal P}_a$ is achieved by a nonnegative function. They also prove that the inequality $S_a \le S_0$ always holds. In particular, if $a \ge 0$ and $a \ne 0$, 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.

MSC:
49J20Optimal control problems with PDE (existence)
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References:
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