# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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:
 49J20 Optimal control problems with PDE (existence)
Full Text:
##### References:
 [1] 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 [2] Benci, V.; Cerami, G.: Existence of positive solutions of the equation-${\Delta}$u + a (x) $u = u(N+2)$/(N-2) in $\mathbb{R}$N. J. fund. Analysis 88, 90-117 (1990) · Zbl 0705.35042 [3] Pan, X.: Positive solutions of the elliptic equation ${\Delta}u + u(N+2)$ (N-2) + K (x) uq = 0 in $\mathbb{R}$N and in balls. J. math. Analysis applic. 172, 323-338 (1993) · Zbl 0801.35025 [4] Azoreno, G.; Alonso, P.: Existence and nonuniqueness for the p-Laplacian nonlinear eigenvalues. Communs partial diff. Eqns 12, 1389-1430 (1987) · Zbl 0637.35069 [5] Brezis, H.; Oswald, L.: Maximization problem involving critical Sobolev exponents. Mat. applic. Comp. 6, 47-56 (1987) · Zbl 0631.49001 [6] Blanchard, P.; Bruning, E.: Variational methods in mathematical physics: a unified approach. (1962) [7] Lions, J.-L.: Séminaire de mathématique supérieures-été 1962. Problèmes aux limites dans LES équations aux dérivées partielles (1965) [8] Willem, M.: Un lemme de déformation quantitatif en calcul des variations. Recherches de mathématique 19. (mai 1992) [9] Lions, P.-L.: The concentration compactness principle in the calculus of variations. The locally compact case, part 2. Ann. inst. H. Poincaré analyse non lineaire 1, 223-283 (1984) · Zbl 0704.49004 [10] Lions, P.-L.: The concentration compactness principle in the calculus of variations. The limit case, part 1. Reva mat. Ibero. 1, 145-201 (1985) · Zbl 0704.49005 [11] Bianchi, G.; Chabrowski, J.; Szulkin, A.: On symmetric solutions of an elliptic equation with a nonlinearity involving critical Sobolev exponent. Nonlinear analysis 25, 41-59 (1995) · Zbl 0823.35051 [12] Hardy, G.H.; Litelwood, J.E.; Pólya, G.: Inequalities. (1934) · Zbl 0010.10703