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)
The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds. (English) Zbl 1161.58310

Let $\left(M,g\right)$ be a smooth compact Riemannian $N$-manifold, $N\ge 2,$ and $p>2$ if $N=2$ and $2 if $N\ge 3·$ The authors show that the positive solutions of the problem

$-{\epsilon }^{2}{{\Delta }}_{g}u+u={u}^{p-1}\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4pt}{0ex}}M$

are generated by stable critical points of the scalar curvature of $g,$ if $\epsilon$ is small enough.

MSC:
 58J05 Elliptic equations on manifolds, general theory 58E30 Variational principles on infinite-dimensional spaces
References:
 [1] Bahri A., Coron J.M.: On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Comm. Pure Appl. Math. 41(3), 253–294 (1988) · Zbl 0649.35033 · doi:10.1002/cpa.3160410302 [2] Benci V., Bonanno C., Micheletti A.M.: On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds. J. Funct. Anal. 252(2), 464–489 (2007) · Zbl 1130.58010 · doi:10.1016/j.jfa.2007.07.010 [3] Brendle, S.: Blow-up phenomena for the Yamabe problem. J. Am. Math. Soc (to appear) [4] Byeon J., Park J.: Singularly perturbed nonlinear elliptic problems on manifolds (English summary). Calc. Var. Partial Differ. Equ. 24(4), 459–477 (2005) · Zbl 1126.58007 · doi:10.1007/s00526-005-0339-4 [5] Del Pino M., Felmer P.L., Wei J.: On the role of mean curvature in some singularly perturbed Neumann problems. SIAM J. Math. Anal. 31(1), 63–79 (1999) · Zbl 0942.35058 · doi:10.1137/S0036141098332834 [6] Floer A., Weinstein A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69(3), 397–408 (1986) · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0 [7] Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in R n . Mathematical analysis and applications. Part A, Adv. Math. Suppl. Stud., vol. 7a, pp. 369–402. Academic Press, New York (1981) [8] Grossi M., Pistoia A.: On the effect of critical points of distance function in superlinear elliptic problems. Adv. Differ. Equ. 5(10–12), 1397–1420 (2000) [9] Grossi M., Pistoia A., Wei J.: Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory. Calc. Var. Partial Differ. Equ. 11(2), 143–175 (2000) · doi:10.1007/PL00009907 [10] Gui C.: Multipeak solutions for a semilinear Neumann problem. Duke Math. J. 84(3), 739–769 (1996) · Zbl 0866.35039 · doi:10.1215/S0012-7094-96-08423-9 [11] Gui C., Wei J.: Multiple interior peak solutions for some singularly perturbed Neumann problems. J. Differ. Equ. 158(1), 1–27 (1999) · Zbl 1061.35502 · doi:10.1016/S0022-0396(99)80016-3 [12] Gui C., Wei J., Winter M.: Multiple boundary peak solutions for some singularly perturbed Neumann problems. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(1), 47–82 (2000) · Zbl 0944.35020 · doi:10.1016/S0294-1449(99)00104-3 [13] Kazdan J.L., Warner F.: Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvature. Ann. Math. 101, 317–331 (1975) · Zbl 0297.53020 · doi:10.2307/1970993 [14] Kwong M.K.: Uniqueness of positive solutions of ${\Delta }$u u + u p = 0 in ${ℝ}^{n}$ . Arch. Ration. Mech. Anal. 105, 243–266 (1989) · Zbl 0676.35032 · doi:10.1007/BF00251502 [15] Hebey E., Vaugon M.: Courbure scalaire prescrite pour des varietes non conformement diffeomorphes a la sphere. C. R. Acad. Sci. Paris, Ser. I 316, 281–282 (1993) [16] Hebey E.: Changements de metriques conformes sur la sphere. Le probleme de Nirenberg. Bull. Sc. Math. 2 e Ser. 114, 215–242 (1990) [17] Hebey E.: Scalar curvature on S N and first spherical harmonics. Differ. Geom. Appl. 5, 71–78 (1995) · Zbl 0816.58039 · doi:10.1016/0926-2245(95)00007-Q [18] Hirano, N.: Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold. Nonlinear Anal. (to appear) [19] Li Y.Y.: On a singularly perturbed ellittic equation. Adv. Differ. Equ. 2(6), 955–580 (1997) [20] Li Y.Y.: On a singularly perturbed equation with Neumann boundary condition. Comm. Partial Differ. Equ. 23(3–4), 487–545 (1998) [21] Lin C.S., Ni W.M., Takagi I.: Large amplitude stationary solutions to a chemotaxis system. J. Differ. Equ. 72(1), 1–27 (1988) · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7 [22] Ni W.M., Takagi I.: On the shape of least-energy solutions to a semilinear Neumann problem. Comm. Pure Appl. Math. 44(7), 819–851 (1991) · Zbl 0754.35042 · doi:10.1002/cpa.3160440705 [23] Ni W.M., Takagi I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70(2), 247–281 (1993) · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4 [24] Visetti, D.: Multiplicity of solutions of a zero-mass nonlinear equation on a Riemannian manifold. J. Differ. Equ. (to appear) [25] Wei J.: On the boundary spike layer solutions to a singularly perturbed Neumann problem. J. Differ. Equ. 134(1), 104–133 (1997) · Zbl 0873.35007 · doi:10.1006/jdeq.1996.3218 [26] Wei J.: On the interior spike layer solutions to a singularly perturbed Neumann problem (English summary). Tohoku Math. J. 50(2), 159–178 (1998) · Zbl 0918.35024 · doi:10.2748/tmj/1178224971 [27] Wei J., Winter M.: Multi-peak solutions for a wide class of singular perturbation problems (English summary). J. Lond. Math. Soc. 59(2), 585–606 (1999) · doi:10.1112/S002461079900719X