Perturbative techniques for the construction of spike-layers. (English) Zbl 1435.35032
Summary: In this paper we survey some results concerning the construction of spike-layers, namely solutions to singularly perturbed equations that exhibit a concentration behaviour. Their study is motivated by the analysis of pattern formation in biological systems such as the Keller-Segel or the Gierer-Meinhardt’s. We describe some general perturbative variational strategy useful to study concentration at points, and also at spheres in radially symmetric situations.
MSC:
| 35B25 | Singular perturbations in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J61 | Semilinear elliptic equations |
| 53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |
Keywords:
singularly perturbed elliptic problems; finite-dimensional reduction; pattern formation; concentration behaviourReferences:
| [1] | A. Ambrosetti; M. Badiale; S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140, 285-300 (1997) · Zbl 0896.35042 |
| [2] | A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic problems on \(\begin{document} \mathbb{R}^N\end{document} \), Birkhäuser, Progr. in Math., 240 (2005). |
| [3] | A. Ambrosetti; A. Malchiodi; W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅰ, Comm. Math. Phys., 235, 427-466 (2003) · Zbl 1072.35019 |
| [4] | A. Ambrosetti; A. Malchiodi; W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅱ, Indiana Univ. Math. J., 53, 297-329 (2004) · Zbl 1081.35008 · doi:10.1512/iumj.2004.53.2400 |
| [5] | W. W. Ao; M. Musso; J. C. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM J. Math. Anal., 43, 2519-2541 (2011) · Zbl 1237.35038 · doi:10.1137/100812100 |
| [6] | W. W. Ao; H. Chan; J. C. Wei, Boundary concentrations on segments for the Lin-Ni-Takagi problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18, 653-696 (2018) · Zbl 1396.35011 |
| [7] | M. Badiale; T. D’Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation, Nonlinear Anal. Ser. A: Theory Methods, 49, 947-985 (2002) · Zbl 1018.35021 · doi:10.1016/S0362-546X(01)00717-9 |
| [8] | T. Bartsch; S. J. Peng, Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations. I, Indiana Univ. Math. J., 57, 1599-1631 (2008) · Zbl 1155.35026 · doi:10.1512/iumj.2008.57.3243 |
| [9] | T. Bartsch; S. J. Peng, Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations. Ⅱ, J. Differential Equations, 248, 2746-2767 (2010) · Zbl 1195.35143 · doi:10.1016/j.jde.2010.02.014 |
| [10] | V. Benci; T. D’Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations, 184, 109-138 (2002) · Zbl 1060.35129 · doi:10.1006/jdeq.2001.4138 |
| [11] | H. Berestycki; P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029 · doi:10.1007/BF00250555 |
| [12] | D. Bonheure, J.-B. Casteras and B. Noris, Multiple positive solutions of the stationary Keller-Segel system, Calc. Var. Partial Differential Equations, 56 (2017), Art. 74, 35 pp. · Zbl 1375.35097 |
| [13] | R. G. Casten; C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Eq., 27, 266-273 (1978) · Zbl 0338.35055 · doi:10.1016/0022-0396(78)90033-5 |
| [14] | E. N. Dancer; S. S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189, 241-262 (1999) · Zbl 0933.35070 · doi:10.2140/pjm.1999.189.241 |
| [15] | J. Davila; A. Pistoia; G. Vaira, Bubbling solutions for supercritical problems on manifolds, J. Math. Pures Appl., 103, 1410-1440 (2015) · Zbl 1347.58004 · doi:10.1016/j.matpur.2014.11.004 |
| [16] | M. del Pino; P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4, 121-137 (1996) · Zbl 0844.35032 · doi:10.1007/BF01189950 |
| [17] | M. del Pino; P. L. Felmer, Semi-classcal states for nonlinear Schröedinger equations, J. Funct. Anal., 149, 245-265 (1997) · Zbl 0887.35058 · doi:10.1006/jfan.1996.3085 |
| [18] | M. del Pino; P. L. Felmer; J. C. Wei, On the role of the mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31, 63-79 (1999) · Zbl 0942.35058 · doi:10.1137/S0036141098332834 |
| [19] | M. del Pino; M. Kowalczyk; J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60, 113-146 (2007) · Zbl 1123.35003 · doi:10.1002/cpa.20135 |
| [20] | M. del Pino; M. Kowalczyk; F. Pacard; J. C. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224, 1462-1516 (2010) · Zbl 1197.35114 · doi:10.1016/j.aim.2010.01.003 |
| [21] | M. del Pino; F. Mahmoudi; M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents, J. Eur. Math. Soc. (JEMS), 16, 1687-1748 (2014) · Zbl 1310.35136 · doi:10.4171/JEMS/473 |
| [22] | A. Floer; A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69, 397-408 (1986) · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0 |
| [23] | B. Gidas; W. M. Ni; L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125 |
| [24] | A. Gierer; H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12, 30-39 (1972) · Zbl 1434.92013 |
| [25] | M. Grossi; A. Pistoia; J. C. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations, 11, 143-175 (2000) · Zbl 0964.35047 · doi:10.1007/PL00009907 |
| [26] | C. F. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84, 739-769 (1996) · Zbl 0866.35039 · doi:10.1215/S0012-7094-96-08423-9 |
| [27] | C. F. Gui; J. C. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52, 522-538 (2000) · Zbl 0949.35052 · doi:10.4153/CJM-2000-024-x |
| [28] | C. F. Gui; J. C. Wei; M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17, 47-82 (2000) · Zbl 0944.35020 · doi:10.1016/S0294-1449(99)00104-3 |
| [29] | Y. Guo; J. Yang, Concentration on surfaces for a singularly perturbed Neumann problem in three-dimensional domains, J. Differential Equations, 255, 2220-2266 (2013) · Zbl 1285.35011 · doi:10.1016/j.jde.2013.06.011 |
| [30] | M. K. Kwong, Uniqueness of positive solutions of \(\begin{document}- \Delta u + u + u^p = 0\end{document}\) in \(\begin{document} \mathbb{R}^N\end{document} \), Arch. Rational Mech. Anal., 105, 243-266 (1989) · Zbl 0676.35032 · doi:10.1007/BF00251502 |
| [31] | Y. Y. Li, On a singularly perturbed equation with Neumann boundary conditions, Comm. Partial Differential Equations, 23, 487-545 (1998) · Zbl 0898.35004 · doi:10.1080/03605309808821354 |
| [32] | Y. Y. Li; L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51, 1445-1490 (1998) · Zbl 0933.35083 · doi:10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z |
| [33] | C.-S. Lin; W.-M. Ni; I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72, 1-27 (1988) · Zbl 0676.35030 · doi:10.1016/0022-0396(88)90147-7 |
| [34] | F.-H. Lin; W.-M. Ni; J.-C. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60, 252-281 (2007) · Zbl 1170.35424 · doi:10.1002/cpa.20139 |
| [35] | F. Mahmoudi; A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209, 460-525 (2007) · Zbl 1160.35011 · doi:10.1016/j.aim.2006.05.014 |
| [36] | F. Mahmoudi; A. Malchiodi; M. Montenegro, Solutions to the nonlinear Schr inger equation carrying momentum along a curve, Comm. Pure Appl. Math., 62, 1155-1264 (2009) · Zbl 1180.35485 |
| [37] | F. Mahmoudi; R. Mazzeo; F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16, 924-958 (2006) · Zbl 1108.53031 · doi:10.1007/s00039-006-0566-7 |
| [38] | F. Mahmoudi; F. Sáchez; W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds, J. Differential Equations, 258, 243-280 (2015) · Zbl 1307.58008 · doi:10.1016/j.jde.2014.09.010 |
| [39] | A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains, Geom. Funct. Anal., 15, 1162-1222 (2005) · Zbl 1087.35010 · doi:10.1007/s00039-005-0542-7 |
| [40] | A. Malchiodi, Construction of multidimensional spike-layers, Discrete Contin. Dyn. Syst., 14, 187-202 (2006) · Zbl 1220.35050 · doi:10.3934/dcds.2006.14.187 |
| [41] | A. Malchiodi, Some new entire solutions of semilinear elliptic equations on \(\begin{document} \mathbb{R}^N\end{document} \), Adv. Math., 221, 1843-1909 (2009) · Zbl 1178.35186 · doi:10.1016/j.aim.2009.03.012 |
| [42] | A. Malchiodi; M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math, 15, 1507-1568 (2002) · Zbl 1124.35305 · doi:10.1002/cpa.10049 |
| [43] | A. Malchiodi; M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124, 105-143 (2004) · Zbl 1065.35037 · doi:10.1215/S0012-7094-04-12414-5 |
| [44] | A. Malchiodi; W.-M. Ni; J.-C. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22, 143-163 (2005) · Zbl 1207.35141 · doi:10.1016/j.anihpc.2004.05.003 |
| [45] | H. Matano, Asymptotic behavior and stability of solutions of semilinear diffusion equations, Publ. Res. Inst. Math. Sci., 15, 401-454 (1979) · Zbl 0445.35063 · doi:10.2977/prims/1195188180 |
| [46] | R. Mazzeo; F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom., 13, 633-670 (2005) · Zbl 1096.53035 · doi:10.4310/CAG.2005.v13.n4.a1 |
| [47] | W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45, 9-18 (1998) · Zbl 0917.35047 |
| [48] | W.-M. Ni; I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297, 351-368 (1986) · Zbl 0635.35031 · doi:10.1090/S0002-9947-1986-0849484-2 |
| [49] | W.-M. Ni; I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., 41, 819-851 (1991) · Zbl 0754.35042 · doi:10.1002/cpa.3160440705 |
| [50] | W. M. Ni; I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70, 247-281 (1993) · Zbl 0796.35056 · doi:10.1215/S0012-7094-93-07004-4 |
| [51] | W.-M. Ni; I. Takagi; E. Yanagida, Stability of least energy patterns of the shadow system for an activator-inhibitor model. Recent topics in mathematics moving toward science and engineering, Japan J. Indust. Appl. Math., 18, 259-272 (2001) · Zbl 1200.35172 · doi:10.1007/BF03168574 |
| [52] | W.-M. Ni; J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48, 731-768 (1995) · Zbl 0838.35009 · doi:10.1002/cpa.3160480704 |
| [53] | Y.-G. Oh, On positive Multi-lump bound states of nonlinear Schrödinger equations under multiple well potentials, Comm. Math. Phys., 131, 223-253 (1990) · Zbl 0753.35097 · doi:10.1007/BF02161413 |
| [54] | S. Santra; J. C. Wei, New entire positive solution for the nonlinear Schräinger equation: Coexistence of fronts and bumps, Amer. J. Math., 135, 443-491 (2013) · Zbl 1307.35126 · doi:10.1353/ajm.2013.0014 |
| [55] | J. P. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc., 354, 3117-3154 (2002) · Zbl 0992.35031 · doi:10.1090/S0002-9947-02-03007-6 |
| [56] | W. A. Strauss, Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028 · doi:10.1007/BF01626517 |
| [57] | A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237, 37-72 (1952) · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012 |
| [58] | L. P. Wang; J. C. Wei; J. Yang, On Ambrosetti-Malchiodi-Ni conjecture for general hypersurfaces, Comm. Partial Differential Equations, 36, 2117-2161 (2011) · Zbl 1235.35265 · doi:10.1080/03605302.2011.580033 |
| [59] | Z. Q. Wang, On the existence of multiple, single-peaked solutions for a semilinear Neumann problem, Arch. Rational Mech. Anal., 120, 375-399 (1992) · Zbl 0784.35035 · doi:10.1007/BF00380322 |
| [60] | J. C. Wei, On the construction of single-peaked solutions to a singularly perturbed semilinear Dirichlet problem, J. Differential Equations, 129, 315-333 (1996) · Zbl 0865.35011 · doi:10.1006/jdeq.1996.0120 |
| [61] | J. C. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134, 104-133 (1997) · Zbl 0873.35007 · doi:10.1006/jdeq.1996.3218 |
| [62] | J. C. Wei; J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains, Indiana Univ. Math. J., 56, 3025-3073 (2007) · Zbl 1173.35063 · doi:10.1512/iumj.2007.56.3133 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
