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Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds. (English) Zbl 1159.58012

Let \((M, g)\) be a smooth, compact, \(N\)-dimensional Riemannian manifold.
The authors prove that for any fixed positive integer \(K\) the problem \[ -\varepsilon^2\Delta_g u +u=u^{p-1}\;\text{in}\;M,\quad u > 0 \;\text{in}\;M \] has a \(K\)-peaks solution. Here \(p > 2\) if \(N = 2\) and \(2 < p < 2^*=\frac{2N}{N-2}\) when \(N\geq 3.\) Moreover, the peaks collapse, as \(\varepsilon\to0,\) to an isolated local minimum point of the scalar curvature.

MSC:

58J05 Elliptic equations on manifolds, general theory
58E30 Variational principles in infinite-dimensional spaces
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[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] Byeon J., Park J.: Singularly perturbed nonlinear elliptic problems on manifolds. Calc. Var. Partial Differ. Equ. 24(4), 459–477 (2005) · Zbl 1126.58007 · doi:10.1007/s00526-005-0339-4
[4] Dancer E.N., Yan S.: Multipeak solutions for a singularly perturbed Neumann problem. Pac. J. Math. 189(2), 241–262 (1999) · Zbl 0933.35070 · doi:10.2140/pjm.1999.189.241
[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] 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
[8] 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
[9] Hirano, N.: Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold. Nonlinear Anal (2008) (in press) · Zbl 1153.35031
[10] Li Y.Y.: On a singularly perturbed equation with Neumann boundary condition. Comm. Partial Differ. Equ. 23(3–4), 487–545 (1998) · Zbl 0898.35004
[11] 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
[12] Micheletti, A.M., Pistoia, A.: The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds. Calculus of Variations PDE (2008) (in press) · Zbl 1161.58310
[13] Micheletti, A.M., Pistoia, A.: Multi-peak solutions for a singularly perturbed nonlinear elliptic problem on Riemannian manifolds (2008) (preprint) · Zbl 1207.35037
[14] 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
[15] 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
[16] 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
[17] Wei J., Winter M.: Multi-peak solutions for a wide class of singular perturbation problems. J. London Math. Soc. 59(2), 585–606 (1999) · Zbl 0922.35025 · doi:10.1112/S002461079900719X
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