. Then, we develop a solvability theory for the linearized operator L defined in (1.22) under suitable orthogonality conditions

, A.4 The reduced energy

, as ? ? 0, J ? (U ? + ? ? ) = J ? (U ? ) + h.o.t. C 1-uniformly on compact sets of (0, +?) × R n (see

, Lemma 1.A.2. Let a, b ? R + be fixed numbers such that

T. Aubin and S. Bismuth, Courbure scalaire prescrite sur les variétés riemanniennes compactes dans le cas négatif, J. Funct. Anal, vol.143, issue.2, pp.529-541, 1997.

S. Alama, L. Bronsard, and J. A. Montero, On the Ginzburg-Landau model of a superconducting ball in a uniform field, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.23, issue.2, p.2201153, 2006.

G. Alberti, S. Baldo, and G. Orlandi, Variational convergence for functionals of GinzburgLandau type, Indiana Univ. Math. J, vol.54, issue.5, p.2177107, 2005.
DOI : 10.1512/iumj.2005.54.2601
URL : http://cvgmt.sns.it/papers/albbalorl02a/abo2.pdf

A. A. Abrikosov, On the Magnetic properties of superconductors of the second group, Sov. Phys. JETP, vol.5, pp.1174-1182, 1957.

A. Aftalion, Vortices in Bose-Einstein condensates, Progress in Nonlinear Differential Equations and their Applications, vol.67, p.2228356, 2006.
DOI : 10.1016/j.crhy.2004.01.001
URL : https://hal.archives-ouvertes.fr/hal-00017351

T. Aubin and E. Hebey, Courbure scalaire prescrite, Bull. Sci. Math, vol.115, issue.2, pp.125-131, 1991.

G. Adimurthi and . Mancini, The Neumann problem for elliptic equations with critical nonlinearity, Nonlinear analysis, p.1205370, 1991.

O. Agudelo and A. Pistoia, Boundary concentration phenomena for the higher-dimensional Keller-Segel system, Calc. Var. Partial Differential Equations, vol.55, issue.6, p.3566211, 2016.

F. Adimurthi, S. L. Pacella, and . Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal, vol.113, issue.2, 1993.

T. Aubin, ´ Equations différentielles non linéaires etprobì eme de Yamabe concernant la courbure scalaire, J. Math. Pures Appl, issue.9, p.431287, 1976.

T. Aubin, Probì emes isopérimétriques et espaces de Sobolev, J. Differential Geometry, vol.11, issue.4, pp.573-598, 1976.

T. Aubin, Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, p.1636569, 1998.

S. L. Adimurthi and . Yadava, Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents, Arch. Rational Mech. Anal, vol.115, issue.3, p.1106295, 1991.

S. L. Adimurthi and . Yadava, Nonexistence of positive radial solutions of a quasilinear Neumann problem with a critical Sobolev exponent, Arch. Rational Mech. Anal, vol.139, issue.3, p.1480241, 1997.

F. Bethuel, H. Brezis, and F. Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, vol.13, p.1269538, 1994.
DOI : 10.1007/978-3-319-66673-0
URL : https://hal.archives-ouvertes.fr/hal-00114093

J. Bourgain, H. Brezis, and P. Mironescu, H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg-Landau equation, Publ. Math. Inst. HautesÉtudesHautes´HautesÉtudes Sci, vol.99, 2004.

F. Bethuel, H. Brezis, and G. Orlandi, Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions, J. Funct. Anal, vol.186, issue.2, 2001.

H. Brezis, J. Coron, and E. H. Lieb, Harmonic maps with defects, Comm. Math. Phys, vol.107, issue.4, pp.649-705, 1986.

D. Bonheure, J. Casteras, and B. Noris, Layered solutions with unbounded mass for the Keller-Segel equation, J. Fixed Point Theory Appl, vol.19, issue.1, p.3625083, 2017.
DOI : 10.1007/s11784-016-0364-2
URL : https://hal.archives-ouvertes.fr/hal-01398930

D. Bonheure, J. Casteras, and B. Noris, Multiple positive solutions of the stationary KellerSegel system, Calc. Var. Partial Differential Equations, vol.56, p.3641921, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01398922

D. Bonheure, J. Casteras, and C. Román, Unbounded mass radial solutions for the KellerSegel equation in the disk
URL : https://hal.archives-ouvertes.fr/hal-01665514

J. Bardeen, L. N. Cooper, and J. R. Schrieffer, Theory of superconductivity, Phys. Rev, vol.108, pp.1175-1204, 1957.

G. Bianchi and H. Egnell, A note on the Sobolev inequality, J. Funct. Anal, vol.100, issue.1, 1991.

D. Bonheure, M. Grossi, B. Noris, and S. Terracini, Multi-layer radial solutions for a supercritical Neumann problem, J. Differential Equations, vol.261, issue.1, pp.455-504, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01182832

F. Borer, L. Galimberti, and M. Struwe, Large" conformal metrics of prescribed Gauss curvature on surfaces of higher genus, Comment. Math. Helv, vol.90, issue.2, pp.407-428, 2015.

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl, vol.8, issue.2, p.92046, 1998.

S. Bismuth, Courbure scalaire prescrite sur une variété riemannienne C ? compacte dans le cas nul, J. Math. Pures Appl, issue.9, p.1645069, 1998.

S. Baldo, R. L. Jerrard, G. Orlandi, and H. M. Soner, Convergence of Ginzburg-Landau functionals in three-dimensional superconductivity, vol.205, p.2960031, 2012.

S. Baldo, R. L. Jerrard, G. Orlandi, and H. M. Soner, Vortex density models for superconductivity and superfluidity, Comm. Math. Phys, vol.318, issue.1, pp.131-171, 2013.
DOI : 10.1007/s00220-012-1629-2
URL : http://arxiv.org/pdf/1112.0293

A. Bahri, Y. Li, and O. Rey, On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations, vol.3, issue.1, p.1384837, 1995.
URL : https://hal.archives-ouvertes.fr/hal-00943456

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math, vol.36, issue.4, pp.437-477, 1983.

F. Bethuel, G. Orlandi, and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. (JEMS), vol.6, issue.1, pp.17-94, 2004.
DOI : 10.4171/jems/2
URL : https://hal.archives-ouvertes.fr/hal-00020106

F. Bethuel and T. Rivì, Vortices for a variational problem related to superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.12, issue.3, p.1340265, 1995.
DOI : 10.1016/s0294-1449(16)30157-3
URL : https://doi.org/10.1016/s0294-1449(16)30157-3

F. Catrina, Proc. Amer. Math. Soc, vol.137, issue.11, p.2529879, 2009.

D. Chiron, Boundary problems for the Ginzburg-Landau equation, Commun. Contemp. Math, vol.7, issue.5, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00809294

C. Chen and C. Lin, Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces, Comm. Pure Appl. Math, vol.55, issue.6, 2002.

S. A. Chang and P. C. Yang, Prescribing Gaussian curvature on S 2, Acta Math, vol.159, issue.3-4, pp.215-259, 1987.

P. G. De-gennes, Superconductivity of Metals and Alloys, Advanced book classics, 1999.

W. Y. Ding and J. Q. Liu, A note on the problem of prescribing Gaussian curvature on surfaces, Trans. Amer. Math. Soc, vol.347, issue.3, p.1257102, 1995.

M. A. Pino, Positive solutions of a semilinear elliptic equation on a compact manifold, Nonlinear Anal, vol.22, issue.11, 1994.

M. Pino, J. Dolbeault, and M. Musso, The Brezis-Nirenberg problem near criticality in dimension 3, J. Math. Pures Appl, issue.9, p.2103187, 2004.

M. , D. Pino, and P. L. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting, Indiana Univ. Math. J, vol.48, issue.3, pp.883-898, 1999.

M. Pino, P. Felmer, and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, vol.16, issue.2, p.1956850, 2003.

M. Pino, P. L. Felmer, and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal, vol.31, issue.1, p.1742305, 1999.

M. Pino, M. Kowalczyk, and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, vol.24, issue.1, p.2157850, 2005.

M. Pino, M. Musso, and A. Pistoia, Super-critical boundary bubbling in a semilinear Neumann problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.22, issue.1, pp.45-82, 2005.

M. Pino, M. Musso, C. Román, and J. Wei, Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3, Accepted for publication in, J. Anal. Math

M. Pino, A. Pistoia, and G. Vaira, Large mass boundary condensation patterns in the stationary Keller-Segel system, J. Differential Equations, vol.261, issue.6, pp.3414-3462, 2016.

M. Pino and C. Román, Large conformal metrics with prescribed sign-changing Gauss curvature, Calc. Var. Partial Differential Equations, vol.54, issue.1, pp.763-789, 2015.

M. Pino and J. Wei, Collapsing steady states of the Keller-Segel system, Nonlinearity, vol.19, issue.3, p.35130, 2006.

O. Druet, Elliptic equations with critical Sobolev exponents in dimension 3, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.19, issue.2, p.1902741, 2002.
DOI : 10.1016/s0294-1449(02)00095-1
URL : https://hal.archives-ouvertes.fr/hal-00194426

O. Druet, F. Robert, and J. Wei, The Lin-Ni's problem for mean convex domains, Mem. Amer. Math. Soc, vol.218, issue.1027, 2012.
DOI : 10.1090/s0065-9266-2011-00646-5
URL : https://hal.archives-ouvertes.fr/hal-01279343

E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math, vol.189, issue.2, p.1696122, 1999.
DOI : 10.2140/pjm.1999.189.241

P. Esposito and A. Pistoia, Blowing-up solutions for the Yamabe equation, Port. Math, vol.71, issue.3-4, pp.249-276, 2014.
DOI : 10.4171/pm/1952

P. Esposito, A. Pistoia, and J. Vétois, The effect of linear perturbations on the Yamabe problem, Math. Ann, vol.358, issue.1-2, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00769040

J. F. Escobar and R. M. Schoen, Conformal metrics with prescribed scalar curvature, Invent. Math, vol.86, issue.2, pp.243-254, 1986.

P. Esposito, EstimationsàEstimationsà l'intérieur pour unprobì eme elliptique semi-linéaire avec nonlinéarité critique, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.24, issue.4, pp.629-644, 2007.

A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal, vol.69, issue.3, pp.397-408, 1986.

C. Gui and N. Ghoussoub, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z, vol.229, issue.3, p.1658569, 1998.

C. Gui and C. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem, J. Reine Angew. Math, vol.546, p.1900999, 2002.

V. L. Ginzburg and L. D. Landau, On the theory of superconductivity, Zh. Eksp. Teor. Fiz, vol.20, pp.546-568, 1950.

A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, vol.12, issue.1, pp.30-39, 1972.

M. Grossi and B. Noris, Positive constrained minimizers for supercritical problems in the ball, Proc. Amer. Math. Soc, vol.140, issue.6, p.2888200, 2012.

M. Grossi, A. Pistoia, and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations, vol.11, issue.2, p.1782991, 2000.

M. Grossi, Asymptotic behaviour of the Kazdan-Warner solution in the annulus, J. Differential Equations, vol.223, issue.1, p.35122, 2006.

P. M. Gruber, Asymptotic estimates for best and stepwise approximation of convex bodies. I, Forum Math, vol.5, pp.281-297, 1993.
DOI : 10.1515/form.1993.5.281

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, vol.158, issue.1, p.1721719, 1999.

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein, vol.105, issue.3, pp.103-165, 2003.

D. Horstmann, From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, Jahresber. Deutsch. Math.-Verein, vol.II, issue.2, p.2073515, 2004.

E. Hebey and M. Vaugon, Meilleures constantes dans le théorème d'inclusion de Sobolev et multiplicité pour lesprobì emes de Nirenberg et Yamabe, Indiana Univ, Math. J, vol.41, issue.2, p.1183349, 1992.

E. Hebey and M. Vaugon, Leprobì eme de Yamabé equivariant, Bull. Sci. Math, vol.117, issue.2, pp.241-286, 1993.

R. L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal, vol.30, issue.4, p.1684723, 1999.
DOI : 10.1137/s0036141097300581

R. Jerrard, A. Montero, and P. Sternberg, Local minimizers of the Ginzburg-Landau energy with magnetic field in three dimensions, Comm. Math. Phys, vol.249, issue.3, p.2084007, 2004.

R. L. Jerrard and H. M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations, vol.14, issue.2, pp.151-191, 2002.

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, vol.26, issue.3, pp.399-415, 1970.

H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J, vol.40, issue.1, p.1101219, 1991.

J. L. Kazdan and F. W. Warner, Curvature functions for compact 2-manifolds, Ann. of Math, issue.2, p.343205, 1974.
DOI : 10.2307/1971012

J. L. Kazdan and F. W. Warner, Scalar curvature and conformal deformation of Riemannian structure, J. Differential Geometry, vol.10, pp.113-134, 1975.
DOI : 10.4310/jdg/1214432678
URL : https://doi.org/10.4310/jdg/1214432678

R. Langevin, Integral geometry from Buffon to geometers of today, Cours Spécialisés, vol.23, p.3469669, 2015.

Y. Y. Li, Prescribing scalar curvature on S n and related problems. I, J. Differential Equations, vol.120, issue.2, 1995.
DOI : 10.1006/jfan.1993.1138
URL : https://doi.org/10.1006/jfan.1993.1138

Y. Li, Prescribing scalar curvature on S n and related problems. II. Existence and compactness, Comm. Pure Appl. Math, vol.49, issue.6, p.1383201, 1996.
DOI : 10.1006/jfan.1993.1138
URL : https://doi.org/10.1006/jfan.1993.1138

Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations, vol.23, issue.3-4, 1998.

C. S. Lin and W. Ni, On the diffusion coefficient of a semilinear Neumann problem, Calculus of variations and partial differential equations, vol.974610, pp.160-174, 1986.

C. Lin, W. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, vol.72, issue.1, pp.1-27, 1988.

F. Lin, W. Ni, and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math, vol.60, issue.2, p.2275329, 2007.

J. M. Lee and T. H. Parker, The Yamabe problem, Bull. Amer. Math. Soc. (N.S.), vol.17, issue.1, pp.37-91, 1987.

F. Lin and T. Rivì, A quantization property for static Ginzburg-Landau vortices, Comm. Pure Appl. Math, vol.54, issue.2, p.1794353, 2001.

F. Lin and T. Rivì, Complex Ginzburg-Landau equations in high dimensions and codimension two area minimizing currents, J. Eur. Math. Soc. (JEMS), issue.3, p.1714735, 1999.

A. M. Micheletti, A. Pistoia, and J. Vétois, Blow-up solutions for asymptotically critical elliptic equations on Riemannian manifolds, Indiana Univ. Math. J, vol.58, issue.4, pp.1719-1746, 2009.
DOI : 10.1512/iumj.2009.58.3633
URL : https://hal.archives-ouvertes.fr/hal-00806352

J. A. Montero, P. Sternberg, and W. P. Ziemer, Local minimizers with vortices in the GinzburgLandau system in three dimensions, vol.57, pp.99-125, 2004.
DOI : 10.1002/cpa.10113

W. Ni, X. B. Pan, and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J, vol.67, issue.1, 1992.

W. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math, vol.44, issue.7, p.1115095, 1991.

W. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J, vol.70, issue.2, pp.247-281, 1993.

T. Ouyang, On the positive solutions of semilinear equations ?u + ?u + hu p = 0 on compact manifolds. II, vol.40, p.1129343, 1991.

A. Pistoia and C. Román, Large conformal metrics with prescribed scalar curvature, J. Differential Equations, vol.263, issue.9, pp.5902-5938, 2017.
DOI : 10.1016/j.jde.2017.07.005
URL : https://hal.archives-ouvertes.fr/hal-01421278

A. Pistoia and G. Vaira, Steady states with unbounded mass of the Keller-Segel system, Proc. Roy. Soc. Edinburgh Sect. A, vol.145, issue.1, pp.203-222, 2015.

A. Rauzy, Courbures scalaires des variétés d'invariant conforme négatif, Trans. Amer. Math. Soc, vol.347, issue.12, pp.4729-4745, 1995.

A. Rauzy, Multiplicité pour unprobì eme de courbure scalaire prescrite, Bull. Sci. Math, vol.120, issue.2, pp.153-194, 1996.

O. Rey, The question of interior blow-up-points for an elliptic Neumann problem: the critical case, J. Math. Pures Appl, issue.9, p.1968337, 2002.

O. Rey, An elliptic Neumann problem with critical nonlinearity in three-dimensional domains, Commun. Contemp. Math, vol.1, issue.3, p.1707889, 1999.
URL : https://hal.archives-ouvertes.fr/hal-00943493

T. Rivì, Line vortices in the U(1)-Higgs model, ESAIM Contrôle Optim. Calc. Var, vol.1, issue.96, p.1394302, 1995.

C. Román, 3D vortex approximation construction and ?-level estimates for the GinzburgLandau functional

C. Román, Global minimizers for the 3D Ginzburg-Landau functional below the first critical field

C. Román, E. Sandier, and S. Serfaty, Global minimizers for the 3D Ginzburg-Landau functional near the first critical field have bounded vorticity

F. Robert and J. Vétois, Sign-changing blow-up for scalar curvature type equations, Comm. Partial Differential Equations, vol.38, issue.8, p.3169751, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01097285

O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity, J. Eur. Math. Soc, vol.7, issue.4, p.2159223, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00935403

E. Sandier, Ginzburg-Landau minimizers from R n+1 to R n and minimal connections, Indiana Univ. Math. J, vol.50, issue.4, p.1889083, 2001.

L. A. Santaló, Integral geometry and geometric probability, Second, Cambridge Mathematical Library, 2004.

E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal, vol.152, issue.2, 1998.

R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom, vol.20, issue.2, pp.479-495, 1984.

R. Schaaf, Stationary solutions of chemotaxis systems, Trans. Amer. Math. Soc, vol.292, issue.2, p.35020, 1985.

S. Serfaty, On a model of rotating superfluids, ESAIM Control Optim. Calc. Var, vol.6, p.1816073, 2001.

S. Serfaty, Local minimizers for the Ginzburg-Landau energy near critical magnetic field. I, Commun. Contemp. Math, vol.1, issue.2, p.1696100, 1999.

E. Sandier and S. Serfaty, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Ann. Inst. H. Poincaré Anal. Non Linéaire, vol.17, issue.1, p.1743433, 2000.

E. Sandier and S. Serfaty, On the energy of type-II superconductors in the mixed phase, Rev. Math. Phys, vol.12, issue.9, p.1794239, 2000.

E. Sandier and S. Serfaty, A rigorous derivation of a free-boundary problem arising in superconductivity, Ann. Sci. ´ Ecole Norm. Sup, issue.4, pp.561-592, 2000.

T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl, vol.10, issue.1, p.35068, 2000.

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal, vol.191, issue.1, p.35155, 2002.

E. Sandier and S. Serfaty, Ginzburg-Landau minimizers near the first critical field have bounded vorticity, Calc. Var. Partial Differential Equations, vol.17, issue.1, pp.17-28, 2003.

E. Sandier and S. Serfaty, A product-estimate for Ginzburg-Landau and corollaries, J. Funct. Anal, vol.211, issue.1, p.2054623, 2004.

E. Sandier and S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications, vol.70, p.2279839, 2007.
DOI : 10.1007/978-0-8176-4550-2

E. Sandier and I. Shafrir, Small energy Ginzburg-Landau minimizers in R 3, J. Funct. Anal, vol.272, issue.9, pp.3946-3964, 2017.
DOI : 10.1016/j.jfa.2017.01.010

M. Struwe, Applications to nonlinear partial differential equations and Hamiltonian systems, Variational methods, Fourth, vol.34, p.2431434, 2008.

G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl, vol.0463908, issue.4, pp.353-372, 1976.
DOI : 10.1007/bf02418013
URL : http://www.math.jhu.edu/~js/Math646/talenti.sobolev.pdf

M. Tinkham, Introduction to superconductivity, Second, 1996.

N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa, issue.3, pp.265-274, 1968.

D. R. Tilley, J. Tilley, . Superfluidity, and T. Superconductivity, , 1990.

J. L. Vázquez and L. Véron, Solutions positives d'´ equations elliptiques semi-linéaires sur des variétés riemanniennes compactes, C. R. Acad. Sci. Paris Sér. I Math, vol.312, issue.11, pp.811-815, 1991.

Z. Wang, Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains, Nonlinear Anal, vol.27, pp.1281-1306, 1996.

G. Wang and J. Wei, Steady state solutions of a reaction-diffusion system modeling chemotaxis, Math. Nachr, vol.233, p.35078, 2002.

J. Wei and M. Winter, Mathematical aspects of pattern formation in biological systems, Applied Mathematical Sciences, vol.189, p.3114654, 2014.

L. Wang, J. Wei, and S. Yan, A Neumann problem with critical exponent in nonconvex domains and Lin-Ni's conjecture, Trans. Amer. Math. Soc, vol.362, issue.9, p.2645043, 2010.

J. Wei and X. Xu, Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in R 3, Pacific J. Math, vol.221, issue.1, p.2194150, 2005.

J. Wei and S. Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth, J. Math. Pures Appl, issue.9, p.2384573, 2007.

H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J, vol.12, 1960.

M. Zhu, Uniqueness results through a priori estimates. I. A three-dimensional Neumann problem, J. Differential Equations, vol.154, issue.2, pp.284-317, 1999.