June  2016, 36(6): 3035-3076. doi: 10.3934/dcds.2016.36.3035

Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents

1. 

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

2. 

Centro de Modelamiento Matemático (CMM), Universidad de Chile, Beauchef 851, Santiago

3. 

Departamento de Matemática, Pontificia Universidad Catolica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago

Received  March 2015 Revised  October 2015 Published  December 2015

In this paper we consider the following problem \begin{eqnarray} \label{abstract} \quad \left\{ \begin{array}{ll}-\Delta u +u= u^{{n-k+2\over n-k-2} \pm\epsilon} & \mbox{ in } \Omega \\ u>0& \mbox{ in }\Omega                                  (0.1)\\ {\partial u\over\partial\nu}=0 & \mbox{ on } \partial\Omega \end{array} \right. \end{eqnarray} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n\ge 7$, $k$ is an integer with $k\ge 1$, and $\epsilon >0$ is a small parameter. Assume there exists a $k$-dimensional closed, embedded, non degenerate minimal submanifold $K$ in $\partial \Omega$. Under a sign condition on a certain weighted avarage of sectional curvatures of $\partial \Omega$ along $K$, we prove the existence of a sequence $\epsilon = \epsilon_j \to 0$ and of solutions $u_\epsilon$ to (0.1) such that $$ |\nabla u_\epsilon |^2 \, \rightharpoonup \, S \delta_K , \quad {\mbox {as}} \quad \epsilon \to 0 $$ in the sense of measure, where $\delta_K$ denotes a Dirac delta along $K$ and $S$ is a universal positive constant.
Citation: Shengbing Deng, Fethi Mahmoudi, Monica Musso. Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3035-3076. doi: 10.3934/dcds.2016.36.3035
References:
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Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity,, A tribute in honour of G. Prodi. Scuola Norm. Sup. Pisa, (1991), 9. Google Scholar

[2]

Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent,, Comm. Partial Differential Equations, 20 (1995), 591. doi: 10.1080/03605309508821110. Google Scholar

[3]

Adimurthi, F. Pacella and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity,, J. Funct. Anal., 113 (1993), 318. doi: 10.1006/jfan.1993.1053. Google Scholar

[4]

Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and L1- estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent,, Diff. Integ. Equ., 8 (1995), 41. Google Scholar

[5]

W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem,, Journal of Differential Equations, 251 (2011), 881. doi: 10.1016/j.jde.2011.05.009. Google Scholar

[6]

W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem,, SIAM J. Math. Anal., 43 (2011), 2519. doi: 10.1137/100812100. Google Scholar

[7]

G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18. doi: 10.1016/0022-1236(91)90099-Q. Google Scholar

[8]

J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473. doi: 10.1016/j.jde.2008.02.024. Google Scholar

[9]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar

[10]

D. Cao and T. Kupper, On the existence of multipeaked solutions to a semilinear Neumann problem,, Duke Math. J., 97 (1999), 261. doi: 10.1215/S0012-7094-99-09712-0. Google Scholar

[11]

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

[12]

M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: 10.1512/iumj.1999.48.1596. Google Scholar

[13]

M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (2000), 63. doi: 10.1137/S0036141098332834. Google Scholar

[14]

M. del Pino, F. 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. doi: 10.4171/JEMS/473. Google Scholar

[15]

M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent,, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553. doi: 10.4171/JEMS/241. Google Scholar

[16]

M. del Pino, M. Musso and A. Pistoia, Supercritical boundary bubbling in a semilinear Neumann problem,, Ann. Inst. H. Poincare Anal. Non-Linearie, 22 (2005), 45. doi: 10.1016/j.anihpc.2004.05.001. Google Scholar

[17]

N. Ghoussoub and C. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent,, Math. Z., 229 (1998), 443. doi: 10.1007/PL00004663. Google Scholar

[18]

N. Ghoussoub, C. Gui and M. Zhu, On a singularly perturbed Neumann problem with the critical exponent,, Comm. Partial Differential Equations, 26 (2001), 1929. doi: 10.1081/PDE-100107812. Google Scholar

[19]

M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory,, Calc. Var. Partial Differential Equations, 11 (2000), 143. doi: 10.1007/PL00009907. Google Scholar

[20]

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

[21]

C. Gui and C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem,, J. Reine Angew. Math., 546 (2002), 201. doi: 10.1515/crll.2002.044. Google Scholar

[22]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems,, J. Differential Equations, 158 (1999), 1. doi: 10.1016/S0022-0396(99)80016-3. Google Scholar

[23]

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

[24]

C.-S. Lin, Locating the peaks of solutions via the maximum principle, I. The Neumann problem,, Comm. Pure Appl. Math., 54 (2001), 1065. doi: 10.1002/cpa.1017. Google Scholar

[25]

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

[26]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Diff. Equat., 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7. Google Scholar

[27]

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

[28]

F. Mahmoudi, A. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve,, Comm. Pure Appl. Math., 62 (2009), 1155. doi: 10.1002/cpa.20290. Google Scholar

[29]

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

[30]

F. Mahmoudi, F. S. Sanchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture in higher dimension,, J. Differential Equations, 258 (2015), 243. doi: 10.1016/j.jde.2014.09.010. Google Scholar

[31]

S. Maier-Paape, K. Schmitt and Z. Q. Wang, On Neumann problems for semilinear elliptic equations with critical nonlinearity existence and symmetry of multi-peaked solutions,, Comm. Partial Differential Equations, 22 (1997), 1493. doi: 10.1080/03605309708821309. Google Scholar

[32]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar

[33]

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

[34]

A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geom. Funct. Anal., 15 (2005), 1162. doi: 10.1007/s00039-005-0542-7. Google Scholar

[35]

R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes,, Comm. Anal. Geom., 13 (2005), 633. doi: 10.4310/CAG.2005.v13.n4.a1. Google Scholar

[36]

M. Musso and J. Yang, Curve like concentration layers for a singularly perturbed nonlinear problem with critical exponents,, Comm. Partial Differential Equations, 39 (2014), 1048. doi: 10.1080/03605302.2013.851215. Google Scholar

[37]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. Google Scholar

[38]

W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary Partial Differential Equations. Handbook Differential Equations, (2004), 157. doi: 10.1016/S1874-5733(04)80005-6. Google Scholar

[39]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819. doi: 10.1002/cpa.3160440705. Google Scholar

[40]

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

[41]

W.-M. Ni, X.-B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1. doi: 10.1215/S0012-7094-92-06701-9. Google Scholar

[42]

S. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f (u) =0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar

[43]

O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains,, Comm. Contemp. Math., 1 (1999), 405. doi: 10.1142/S0219199799000158. Google Scholar

[44]

O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity,, J. Eur. Math. Soc. (JEMS), 7 (2005), 449. doi: 10.4171/JEMS/35. Google Scholar

[45]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl. (IV), 110 (1976), 353. doi: 10.1007/BF02418013. Google Scholar

[46]

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., 362 (2010), 4581. doi: 10.1090/S0002-9947-10-04955-X. Google Scholar

[47]

Z. Q. Wang, High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent,, Proc. Roy. Soc. Edimburgh, 125 (1995), 1003. doi: 10.1017/S0308210500022617. Google Scholar

[48]

X.-J. Wang, Neumann problem of semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equations, 93 (1991), 283. doi: 10.1016/0022-0396(91)90014-Z. Google Scholar

[49]

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

show all references

References:
[1]

Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity,, A tribute in honour of G. Prodi. Scuola Norm. Sup. Pisa, (1991), 9. Google Scholar

[2]

Adimurthi, G. Mancini and S. L. Yadava, The role of the mean curvature in semilinear Neumann problem involving critical exponent,, Comm. Partial Differential Equations, 20 (1995), 591. doi: 10.1080/03605309508821110. Google Scholar

[3]

Adimurthi, F. Pacella and S. L. Yadava, Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity,, J. Funct. Anal., 113 (1993), 318. doi: 10.1006/jfan.1993.1053. Google Scholar

[4]

Adimurthi, F. Pacella and S. L. Yadava, Characterization of concentration points and L1- estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent,, Diff. Integ. Equ., 8 (1995), 41. Google Scholar

[5]

W. Ao, M. Musso and J. Wei, On spikes concentrating on line-segments to a semilinear Neumann problem,, Journal of Differential Equations, 251 (2011), 881. doi: 10.1016/j.jde.2011.05.009. Google Scholar

[6]

W. Ao, M. Musso and J. Wei, Triple junction solutions for a singularly perturbed Neumann problem,, SIAM J. Math. Anal., 43 (2011), 2519. doi: 10.1137/100812100. Google Scholar

[7]

G. Bianchi and H. Egnell, A note on the Sobolev inequality,, J. Funct. Anal., 100 (1991), 18. doi: 10.1016/0022-1236(91)90099-Q. Google Scholar

[8]

J. Byeon, Singularly perturbed nonlinear Neumann problems with a general nonlinearity,, J. Differential Equations, 244 (2008), 2473. doi: 10.1016/j.jde.2008.02.024. Google Scholar

[9]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, Comm. Pure Appl. Math., 42 (1989), 271. doi: 10.1002/cpa.3160420304. Google Scholar

[10]

D. Cao and T. Kupper, On the existence of multipeaked solutions to a semilinear Neumann problem,, Duke Math. J., 97 (1999), 261. doi: 10.1215/S0012-7094-99-09712-0. Google Scholar

[11]

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

[12]

M. del Pino and P. Felmer, Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting,, Indiana Univ. Math. J., 48 (1999), 883. doi: 10.1512/iumj.1999.48.1596. Google Scholar

[13]

M. del Pino, P. Felmer and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems,, SIAM J. Math. Anal., 31 (2000), 63. doi: 10.1137/S0036141098332834. Google Scholar

[14]

M. del Pino, F. 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. doi: 10.4171/JEMS/473. Google Scholar

[15]

M. del Pino, M. Musso and F. Pacard, Bubbling along boundary geodesics near the second critical exponent,, J. Eur. Math. Soc. (JEMS), 12 (2010), 1553. doi: 10.4171/JEMS/241. Google Scholar

[16]

M. del Pino, M. Musso and A. Pistoia, Supercritical boundary bubbling in a semilinear Neumann problem,, Ann. Inst. H. Poincare Anal. Non-Linearie, 22 (2005), 45. doi: 10.1016/j.anihpc.2004.05.001. Google Scholar

[17]

N. Ghoussoub and C. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent,, Math. Z., 229 (1998), 443. doi: 10.1007/PL00004663. Google Scholar

[18]

N. Ghoussoub, C. Gui and M. Zhu, On a singularly perturbed Neumann problem with the critical exponent,, Comm. Partial Differential Equations, 26 (2001), 1929. doi: 10.1081/PDE-100107812. Google Scholar

[19]

M. Grossi, A. Pistoia and J. Wei, Existence of multipeak solutions for a semilinear elliptic problem via nonsmooth critical point theory,, Calc. Var. Partial Differential Equations, 11 (2000), 143. doi: 10.1007/PL00009907. Google Scholar

[20]

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

[21]

C. Gui and C.-S. Lin, Estimates for boundary-bubbling solutions to an elliptic Neumann problem,, J. Reine Angew. Math., 546 (2002), 201. doi: 10.1515/crll.2002.044. Google Scholar

[22]

C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems,, J. Differential Equations, 158 (1999), 1. doi: 10.1016/S0022-0396(99)80016-3. Google Scholar

[23]

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

[24]

C.-S. Lin, Locating the peaks of solutions via the maximum principle, I. The Neumann problem,, Comm. Pure Appl. Math., 54 (2001), 1065. doi: 10.1002/cpa.1017. Google Scholar

[25]

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

[26]

C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Diff. Equat., 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7. Google Scholar

[27]

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

[28]

F. Mahmoudi, A. Malchiodi and M. Montenegro, Solutions to the nonlinear Schrödinger equation carrying momentum along a curve,, Comm. Pure Appl. Math., 62 (2009), 1155. doi: 10.1002/cpa.20290. Google Scholar

[29]

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

[30]

F. Mahmoudi, F. S. Sanchez and W. Yao, On the Ambrosetti-Malchiodi-Ni conjecture in higher dimension,, J. Differential Equations, 258 (2015), 243. doi: 10.1016/j.jde.2014.09.010. Google Scholar

[31]

S. Maier-Paape, K. Schmitt and Z. Q. Wang, On Neumann problems for semilinear elliptic equations with critical nonlinearity existence and symmetry of multi-peaked solutions,, Comm. Partial Differential Equations, 22 (1997), 1493. doi: 10.1080/03605309708821309. Google Scholar

[32]

A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem,, Comm. Pure Appl. Math., 55 (2002), 1507. doi: 10.1002/cpa.10049. Google Scholar

[33]

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

[34]

A. Malchiodi, Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains,, Geom. Funct. Anal., 15 (2005), 1162. doi: 10.1007/s00039-005-0542-7. Google Scholar

[35]

R. Mazzeo and F. Pacard, Foliations by constant mean curvature tubes,, Comm. Anal. Geom., 13 (2005), 633. doi: 10.4310/CAG.2005.v13.n4.a1. Google Scholar

[36]

M. Musso and J. Yang, Curve like concentration layers for a singularly perturbed nonlinear problem with critical exponents,, Comm. Partial Differential Equations, 39 (2014), 1048. doi: 10.1080/03605302.2013.851215. Google Scholar

[37]

W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. Google Scholar

[38]

W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary Partial Differential Equations. Handbook Differential Equations, (2004), 157. doi: 10.1016/S1874-5733(04)80005-6. Google Scholar

[39]

W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem,, Comm. Pure Appl. Math., 44 (1991), 819. doi: 10.1002/cpa.3160440705. Google Scholar

[40]

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

[41]

W.-M. Ni, X.-B. Pan and I. Takagi, Singular behavior of least-energy solutions of a semi-linear Neumann problem involving critical Sobolev exponents,, Duke Math. J., 67 (1992), 1. doi: 10.1215/S0012-7094-92-06701-9. Google Scholar

[42]

S. Pohozaev, Eigenfunctions of the equation $\Delta u +\lambda f (u) =0$,, Dokl. Akad. Nauk SSSR, 165 (1965), 36. Google Scholar

[43]

O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains,, Comm. Contemp. Math., 1 (1999), 405. doi: 10.1142/S0219199799000158. Google Scholar

[44]

O. Rey and J. Wei, Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity,, J. Eur. Math. Soc. (JEMS), 7 (2005), 449. doi: 10.4171/JEMS/35. Google Scholar

[45]

G. Talenti, Best constant in Sobolev inequality,, Ann. Mat. Pura Appl. (IV), 110 (1976), 353. doi: 10.1007/BF02418013. Google Scholar

[46]

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., 362 (2010), 4581. doi: 10.1090/S0002-9947-10-04955-X. Google Scholar

[47]

Z. Q. Wang, High energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponent,, Proc. Roy. Soc. Edimburgh, 125 (1995), 1003. doi: 10.1017/S0308210500022617. Google Scholar

[48]

X.-J. Wang, Neumann problem of semilinear elliptic equations involving critical Sobolev exponents,, J. Differential Equations, 93 (1991), 283. doi: 10.1016/0022-0396(91)90014-Z. Google Scholar

[49]

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

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