June  2020, 40(6): 3767-3787. doi: 10.3934/dcds.2020055

Perturbative techniques for the construction of spike-layers

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

Dedicated to Wei-Ming Ni with admiration

Received  May 2019 Published  October 2019

Fund Project: Andrea Malchiodi has been supported by the project Geometric Variational Problems and Finanziamento a supporto della ricerca di base from Scuola Normale Superiore and by MIUR Bando PRIN 2015 2015KB9WPT001. He is a member of GNAMPA as part of INdAM

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.

Citation: Andrea Malchiodi. Perturbative techniques for the construction of spike-layers. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3767-3787. doi: 10.3934/dcds.2020055
References:
[1]

A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. 

[2]

A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic problems on $ \mathbb{R}^N$, Birkhäuser, Progr. in Math., 240 (2005).

[3]

A. AmbrosettiA. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅰ, Comm. Math. Phys., 235 (2003), 427-466. 

[4]

A. AmbrosettiA. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅱ, Indiana Univ. Math. J., 53 (2004), 297-329.  doi: 10.1512/iumj.2004.53.2400.

[5]

W. W. AoM. Musso and J. C. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM J. Math. Anal., 43 (2011), 2519-2541.  doi: 10.1137/100812100.

[6]

W. W. AoH. Chan and J. C. Wei, Boundary concentrations on segments for the Lin-Ni-Takagi problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 653-696. 

[7]

M. Badiale and T. D'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 947-985.  doi: 10.1016/S0362-546X(01)00717-9.

[8]

T. Bartsch and S. J. Peng, Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations. I, Indiana Univ. Math. J., 57 (2008), 1599-1631.  doi: 10.1512/iumj.2008.57.3243.

[9]

T. Bartsch and S. J. Peng, Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations. Ⅱ, J. Differential Equations, 248 (2010), 2746-2767.  doi: 10.1016/j.jde.2010.02.014.

[10]

V. Benci and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations, 184 (2002), 109-138.  doi: 10.1006/jdeq.2001.4138.

[11]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  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. doi: 10.1007/s00526-017-1163-3.

[13]

R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Eq., 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5.

[14]

E. N. Dancer and S. S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.  doi: 10.2140/pjm.1999.189.241.

[15]

J. DavilaA. Pistoia and G. Vaira, Bubbling solutions for supercritical problems on manifolds, J. Math. Pures Appl., 103 (2015), 1410-1440.  doi: 10.1016/j.matpur.2014.11.004.

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M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.

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M. del Pino and P. L. Felmer, Semi-classcal states for nonlinear Schröedinger equations, J. Funct. Anal., 149 (1997), 245-265.  doi: 10.1006/jfan.1996.3085.

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M. del PinoP. L. Felmer and J. C. Wei, On the role of the mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79.  doi: 10.1137/S0036141098332834.

[19]

M. del PinoM. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.

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M. del PinoM. KowalczykF. Pacard and J. C. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.  doi: 10.1016/j.aim.2010.01.003.

[21]

M. del PinoF. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents, J. Eur. Math. Soc. (JEMS), 16 (2014), 1687-1748.  doi: 10.4171/JEMS/473.

[22]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.

[23]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[24]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39. 

[25]

M. GrossiA. Pistoia and J. C. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations, 11 (2000), 143-175.  doi: 10.1007/PL00009907.

[26]

C. F. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.  doi: 10.1215/S0012-7094-96-08423-9.

[27]

C. F. Gui and J. C. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.  doi: 10.4153/CJM-2000-024-x.

[28]

C. F. GuiJ. C. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82.  doi: 10.1016/S0294-1449(99)00104-3.

[29]

Y. Guo and J. Yang, Concentration on surfaces for a singularly perturbed Neumann problem in three-dimensional domains, J. Differential Equations, 255 (2013), 2220-2266.  doi: 10.1016/j.jde.2013.06.011.

[30]

M. K. Kwong, Uniqueness of positive solutions of $- \Delta u + u + u^p = 0$ in $ \mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[31]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary conditions, Comm. Partial Differential Equations, 23 (1998), 487-545.  doi: 10.1080/03605309808821354.

[32]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z.

[33]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[34]

F.-H. LinW.-M. Ni and J.-C. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.

[35]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525.  doi: 10.1016/j.aim.2006.05.014.

[36]

F. MahmoudiA. Malchiodi and M. Montenegro, Solutions to the nonlinear Schr inger equation carrying momentum along a curve, Comm. Pure Appl. Math., 62 (2009), 1155-1264. 

[37]

F. MahmoudiR. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16 (2006), 924-958.  doi: 10.1007/s00039-006-0566-7.

[38]

F. MahmoudiF. Sáchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds, J. Differential Equations, 258 (2015), 243-280.  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 (2005), 1162-1222.  doi: 10.1007/s00039-005-0542-7.

[40]

A. Malchiodi, Construction of multidimensional spike-layers, Discrete Contin. Dyn. Syst., 14 (2006), 187-202.  doi: 10.3934/dcds.2006.14.187.

[41]

A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $ \mathbb{R}^N$, Adv. Math., 221 (2009), 1843-1909.  doi: 10.1016/j.aim.2009.03.012.

[42]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math, 15 (2002), 1507-1568.  doi: 10.1002/cpa.10049.

[43]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.  doi: 10.1215/S0012-7094-04-12414-5.

[44]

A. MalchiodiW.-M. Ni and J.-C. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 143-163.  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 (1979), 401-454.  doi: 10.2977/prims/1195188180.

[46]

R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom., 13 (2005), 633-670.  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 (1998), 9-18. 

[48]

W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368.  doi: 10.1090/S0002-9947-1986-0849484-2.

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W.-M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., 41 (1991), 819-851.  doi: 10.1002/cpa.3160440705.

[50]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[51]

W.-M. NiI. Takagi and 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 (2001), 259-272.  doi: 10.1007/BF03168574.

[52]

W.-M. Ni and J. C. Wei, On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems, Comm. Pure Appl. Math., 48 (1995), 731-768.  doi: 10.1002/cpa.3160480704.

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J. C. Wei, On the boundary spike layer solutions of a singularly perturbed semilinear Neumann problem, J. Differential Equations, 134 (1997), 104-133.  doi: 10.1006/jdeq.1996.3218.

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J. C. Wei and J. Yang, Concentration on lines for a singularly perturbed Neumann problem in two-dimensional domains, Indiana Univ. Math. J., 56 (2007), 3025-3073.  doi: 10.1512/iumj.2007.56.3133.

show all references

Dedicated to Wei-Ming Ni with admiration

References:
[1]

A. AmbrosettiM. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal., 140 (1997), 285-300. 

[2]

A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic problems on $ \mathbb{R}^N$, Birkhäuser, Progr. in Math., 240 (2005).

[3]

A. AmbrosettiA. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅰ, Comm. Math. Phys., 235 (2003), 427-466. 

[4]

A. AmbrosettiA. Malchiodi and W. M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres. Ⅱ, Indiana Univ. Math. J., 53 (2004), 297-329.  doi: 10.1512/iumj.2004.53.2400.

[5]

W. W. AoM. Musso and J. C. Wei, Triple junction solutions for a singularly perturbed Neumann problem, SIAM J. Math. Anal., 43 (2011), 2519-2541.  doi: 10.1137/100812100.

[6]

W. W. AoH. Chan and J. C. Wei, Boundary concentrations on segments for the Lin-Ni-Takagi problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 653-696. 

[7]

M. Badiale and T. D'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation, Nonlinear Anal. Ser. A: Theory Methods, 49 (2002), 947-985.  doi: 10.1016/S0362-546X(01)00717-9.

[8]

T. Bartsch and S. J. Peng, Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations. I, Indiana Univ. Math. J., 57 (2008), 1599-1631.  doi: 10.1512/iumj.2008.57.3243.

[9]

T. Bartsch and S. J. Peng, Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations. Ⅱ, J. Differential Equations, 248 (2010), 2746-2767.  doi: 10.1016/j.jde.2010.02.014.

[10]

V. Benci and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations, 184 (2002), 109-138.  doi: 10.1006/jdeq.2001.4138.

[11]

H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345.  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. doi: 10.1007/s00526-017-1163-3.

[13]

R. G. Casten and C. J. Holland, Instability results for reaction diffusion equations with Neumann boundary conditions, J. Diff. Eq., 27 (1978), 266-273.  doi: 10.1016/0022-0396(78)90033-5.

[14]

E. N. Dancer and S. S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math., 189 (1999), 241-262.  doi: 10.2140/pjm.1999.189.241.

[15]

J. DavilaA. Pistoia and G. Vaira, Bubbling solutions for supercritical problems on manifolds, J. Math. Pures Appl., 103 (2015), 1410-1440.  doi: 10.1016/j.matpur.2014.11.004.

[16]

M. del Pino and P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations, 4 (1996), 121-137.  doi: 10.1007/BF01189950.

[17]

M. del Pino and P. L. Felmer, Semi-classcal states for nonlinear Schröedinger equations, J. Funct. Anal., 149 (1997), 245-265.  doi: 10.1006/jfan.1996.3085.

[18]

M. del PinoP. L. Felmer and J. C. Wei, On the role of the mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal., 31 (1999), 63-79.  doi: 10.1137/S0036141098332834.

[19]

M. del PinoM. Kowalczyk and J.-C. Wei, Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., 60 (2007), 113-146.  doi: 10.1002/cpa.20135.

[20]

M. del PinoM. KowalczykF. Pacard and J. C. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Adv. Math., 224 (2010), 1462-1516.  doi: 10.1016/j.aim.2010.01.003.

[21]

M. del PinoF. Mahmoudi and M. Musso, Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents, J. Eur. Math. Soc. (JEMS), 16 (2014), 1687-1748.  doi: 10.4171/JEMS/473.

[22]

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69 (1986), 397-408.  doi: 10.1016/0022-1236(86)90096-0.

[23]

B. GidasW. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  doi: 10.1007/BF01221125.

[24]

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin), 12 (1972), 30-39. 

[25]

M. GrossiA. Pistoia and J. C. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations, 11 (2000), 143-175.  doi: 10.1007/PL00009907.

[26]

C. F. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769.  doi: 10.1215/S0012-7094-96-08423-9.

[27]

C. F. Gui and J. C. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math., 52 (2000), 522-538.  doi: 10.4153/CJM-2000-024-x.

[28]

C. F. GuiJ. C. Wei and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 47-82.  doi: 10.1016/S0294-1449(99)00104-3.

[29]

Y. Guo and J. Yang, Concentration on surfaces for a singularly perturbed Neumann problem in three-dimensional domains, J. Differential Equations, 255 (2013), 2220-2266.  doi: 10.1016/j.jde.2013.06.011.

[30]

M. K. Kwong, Uniqueness of positive solutions of $- \Delta u + u + u^p = 0$ in $ \mathbb{R}^N$, Arch. Rational Mech. Anal., 105 (1989), 243-266.  doi: 10.1007/BF00251502.

[31]

Y. Y. Li, On a singularly perturbed equation with Neumann boundary conditions, Comm. Partial Differential Equations, 23 (1998), 487-545.  doi: 10.1080/03605309808821354.

[32]

Y. Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math., 51 (1998), 1445-1490.  doi: 10.1002/(SICI)1097-0312(199811/12)51:11/12<1445::AID-CPA9>3.0.CO;2-Z.

[33]

C.-S. LinW.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis systems, J. Differential Equations, 72 (1988), 1-27.  doi: 10.1016/0022-0396(88)90147-7.

[34]

F.-H. LinW.-M. Ni and J.-C. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.

[35]

F. Mahmoudi and A. Malchiodi, Concentration on minimal submanifolds for a singularly perturbed Neumann problem, Adv. Math., 209 (2007), 460-525.  doi: 10.1016/j.aim.2006.05.014.

[36]

F. MahmoudiA. Malchiodi and M. Montenegro, Solutions to the nonlinear Schr inger equation carrying momentum along a curve, Comm. Pure Appl. Math., 62 (2009), 1155-1264. 

[37]

F. MahmoudiR. Mazzeo and F. Pacard, Constant mean curvature hypersurfaces condensing on a submanifold, Geom. Funct. Anal., 16 (2006), 924-958.  doi: 10.1007/s00039-006-0566-7.

[38]

F. MahmoudiF. Sáchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture for general submanifolds, J. Differential Equations, 258 (2015), 243-280.  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 (2005), 1162-1222.  doi: 10.1007/s00039-005-0542-7.

[40]

A. Malchiodi, Construction of multidimensional spike-layers, Discrete Contin. Dyn. Syst., 14 (2006), 187-202.  doi: 10.3934/dcds.2006.14.187.

[41]

A. Malchiodi, Some new entire solutions of semilinear elliptic equations on $ \mathbb{R}^N$, Adv. Math., 221 (2009), 1843-1909.  doi: 10.1016/j.aim.2009.03.012.

[42]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math, 15 (2002), 1507-1568.  doi: 10.1002/cpa.10049.

[43]

A. Malchiodi and M. Montenegro, Multidimensional boundary layers for a singularly perturbed Neumann problem, Duke Math. J., 124 (2004), 105-143.  doi: 10.1215/S0012-7094-04-12414-5.

[44]

A. MalchiodiW.-M. Ni and J.-C. Wei, Multiple clustered layer solutions for semilinear Neumann problems on a ball, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 143-163.  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 (1979), 401-454.  doi: 10.2977/prims/1195188180.

[46]

R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes, Comm. Anal. Geom., 13 (2005), 633-670.  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 (1998), 9-18. 

[48]

W.-M. Ni and I. Takagi, On the Neumann problem for some semilinear elliptic equations and systems of activator-inhibitor type, Trans. Amer. Math. Soc., 297 (1986), 351-368.  doi: 10.1090/S0002-9947-1986-0849484-2.

[49]

W.-M. Ni and I. Takagi, On the shape of least-energy solution to a semilinear Neumann problem, Comm. Pure Appl. Math., 41 (1991), 819-851.  doi: 10.1002/cpa.3160440705.

[50]

W. M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281.  doi: 10.1215/S0012-7094-93-07004-4.

[51]

W.-M. NiI. Takagi and 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 (2001), 259-272.  doi: 10.1007/BF03168574.

[52]

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