American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity
  • DCDS Home
  • This Issue
  • Next Article
    Dimension estimates for arbitrary subsets of limit sets of a Markov construction and related multifractal analysis
May  2014, 34(5): 2333-2357. doi: 10.3934/dcds.2014.34.2333

Solutions with clustered bubbles and a boundary layer of an elliptic problem

Liping Wang 1, and  Chunyi Zhao 2,

1. 

Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241
2. 

Department of Mathematics, East China Normal University, Shanghai 200241

Received  June 2013 Revised  August 2013 Published  October 2013

We study positive solutions of the equation $ε^2 \Delta u - u + u^\frac{n+2}{n-2} = 0$ where $ε >0$ is small, with Neumann boundary condition in a unit ball $B\subset\mathbb R^3$. We prove the existence of solutions with multiple interior bubbles near the center and a boundary layer. The method may also be used to the case $n=4$, $5$ and get the analogous results.
Keywords: Lyapunov-Schmidt reduction., blow up, Clustered interior bubbles, boundary layer.
Mathematics Subject Classification: 35J25, 35J6.
Citation: Liping Wang, Chunyi Zhao. Solutions with clustered bubbles and a boundary layer of an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2333-2357. doi: 10.3934/dcds.2014.34.2333
References:
[1]

Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity,, Nonlinear Anal. Scuola Norm. Sup. Pisa, 4 (1991), 9.   Google Scholar

[2]

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

[3]

W. W. Ao, J. C. Wei and J. Zeng, An optimal bound on the number of interior spike solutions of the Lin-Ni-Takagi problem,, J. Funct. Anal., 265 (2013), 1324.  doi: 10.1016/j.jfa.2013.06.016.  Google Scholar

[4]

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

[5]

D. M. Cao, E. Noussair and S. S. Yan, Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem,, Pacific J. Math., 200 (2001), 19.  doi: 10.2140/pjm.2001.200.19.  Google Scholar

[6]

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

[7]

P. Esposito, Estimations à l'intérieur pour un problème elliptique semi-linéaire avec non-linéarité critique, [Interior estimates for some semilinear elliptic problem with critical nonlinearity],, Ann. Inst. H. Poincar Anal. Non Linéaire, 24 (2007), 629.  doi: 10.1016/j.anihpc.2006.04.004.  Google Scholar

[8]

N. Ghoussoub and C. F. 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

[9]

N. Ghoussoub, C. F. Gui and M. J. 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

[10]

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

[11]

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

[12]

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

[13]

W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary partial differential equations, I (2004), 157.  doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

[14]

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

[15]

O. Rey, The role of the Green's function in a nonlinear elliptic problem involving the critical Sobolev exponent,, J. Funct. Anal., 89 (1990), 1.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[16]

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

[17]

O. Rey, The question of interior blow-up points for an elliptic Neumann problem: the critical case,, J. Math. Pures Appl., 81 (2002), 655.  doi: 10.1016/S0021-7824(01)01251-X.  Google Scholar

[18]

L. P. Wang and J. C. Wei, Solutions with interior bubble and boundary layer for an elliptic problem,, Discrete Contin. Dyn. Syst., 21 (2008), 333.  doi: 10.3934/dcds.2008.21.333.  Google Scholar

[19]

J. C. Wei, Existence and stability of spikes for the Gierer-Meinhardt system,, in Handbook of Differential Equations: Ationary Partial Differential Equations, V (2008), 487.  doi: 10.1016/S1874-5733(08)80013-7.  Google Scholar

[20]

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

[21]

J. C. Wei and S. S. Yan, Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity,, C. R. Acad. Sci. Paris, 343 (2006), 311.  doi: 10.1016/j.crma.2006.07.010.  Google Scholar

[22]

J. C. Wei and S. S. Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth,, J. Math. Pures Appl. (9), 88 (2007), 350.  doi: 10.1016/j.matpur.2007.07.001.  Google Scholar

show all references

References:
[1]

Adimurthi and G. Mancini, The Neumann problem for elliptic equations with critical nonlinearity,, Nonlinear Anal. Scuola Norm. Sup. Pisa, 4 (1991), 9.   Google Scholar

[2]

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

[3]

W. W. Ao, J. C. Wei and J. Zeng, An optimal bound on the number of interior spike solutions of the Lin-Ni-Takagi problem,, J. Funct. Anal., 265 (2013), 1324.  doi: 10.1016/j.jfa.2013.06.016.  Google Scholar

[4]

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

[5]

D. M. Cao, E. Noussair and S. S. Yan, Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem,, Pacific J. Math., 200 (2001), 19.  doi: 10.2140/pjm.2001.200.19.  Google Scholar

[6]

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

[7]

P. Esposito, Estimations à l'intérieur pour un problème elliptique semi-linéaire avec non-linéarité critique, [Interior estimates for some semilinear elliptic problem with critical nonlinearity],, Ann. Inst. H. Poincar Anal. Non Linéaire, 24 (2007), 629.  doi: 10.1016/j.anihpc.2006.04.004.  Google Scholar

[8]

N. Ghoussoub and C. F. 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

[9]

N. Ghoussoub, C. F. Gui and M. J. 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

[10]

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

[11]

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

[12]

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

[13]

W.-M. Ni, Qualitative properties of solutions to elliptic problems,, in Stationary partial differential equations, I (2004), 157.  doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

[14]

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

[15]

O. Rey, The role of the Green's function in a nonlinear elliptic problem involving the critical Sobolev exponent,, J. Funct. Anal., 89 (1990), 1.  doi: 10.1016/0022-1236(90)90002-3.  Google Scholar

[16]

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

[17]

O. Rey, The question of interior blow-up points for an elliptic Neumann problem: the critical case,, J. Math. Pures Appl., 81 (2002), 655.  doi: 10.1016/S0021-7824(01)01251-X.  Google Scholar

[18]

L. P. Wang and J. C. Wei, Solutions with interior bubble and boundary layer for an elliptic problem,, Discrete Contin. Dyn. Syst., 21 (2008), 333.  doi: 10.3934/dcds.2008.21.333.  Google Scholar

[19]

J. C. Wei, Existence and stability of spikes for the Gierer-Meinhardt system,, in Handbook of Differential Equations: Ationary Partial Differential Equations, V (2008), 487.  doi: 10.1016/S1874-5733(08)80013-7.  Google Scholar

[20]

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

[21]

J. C. Wei and S. S. Yan, Solutions with interior bubble and boundary layer for an elliptic Neumann problem with critical nonlinearity,, C. R. Acad. Sci. Paris, 343 (2006), 311.  doi: 10.1016/j.crma.2006.07.010.  Google Scholar

[22]

J. C. Wei and S. S. Yan, Arbitrary many boundary peak solutions for an elliptic Neumann problem with critical growth,, J. Math. Pures Appl. (9), 88 (2007), 350.  doi: 10.1016/j.matpur.2007.07.001.  Google Scholar

[1]

Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201
[2]

Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739
[3]

Yihong Du, Zongming Guo, Feng Zhou. Boundary blow-up solutions with interior layers and spikes in a bistable problem. Discrete & Continuous Dynamical Systems - A, 2007, 19 (2) : 271-298. doi: 10.3934/dcds.2007.19.271
[4]

Liping Wang, Juncheng Wei. Solutions with interior bubble and boundary layer for an elliptic problem. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 333-351. doi: 10.3934/dcds.2008.21.333
[5]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449
[6]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267
[7]

Bernard Brighi, Tewfik Sari. Blowing-up coordinates for a similarity boundary layer equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 929-948. doi: 10.3934/dcds.2005.12.929
[8]

Huyuan Chen, Hichem Hajaiej, Ying Wang. Boundary blow-up solutions to fractional elliptic equations in a measure framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1881-1903. doi: 10.3934/dcds.2016.36.1881
[9]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71
[10]

Pavol Quittner, Philippe Souplet. Blow-up rate of solutions of parabolic poblems with nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 671-681. doi: 10.3934/dcdss.2012.5.671
[11]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[12]

Françoise Demengel, O. Goubet. Existence of boundary blow up solutions for singular or degenerate fully nonlinear equations. Communications on Pure & Applied Analysis, 2013, 12 (2) : 621-645. doi: 10.3934/cpaa.2013.12.621
[13]

Yihong Du, Zongming Guo. The degenerate logistic model and a singularly mixed boundary blow-up problem. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 1-29. doi: 10.3934/dcds.2006.14.1
[14]

Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54
[15]

Jun Yang, Xiaolin Yang. Clustered interior phase transition layers for an inhomogeneous Allen-Cahn equation in higher dimensional domains. Communications on Pure & Applied Analysis, 2013, 12 (1) : 303-340. doi: 10.3934/cpaa.2013.12.303
[16]

Antoine Sellier. Boundary element approach for the slow viscous migration of spherical bubbles. Discrete & Continuous Dynamical Systems - B, 2011, 15 (4) : 1045-1064. doi: 10.3934/dcdsb.2011.15.1045
[17]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521
[18]

C. Brändle, F. Quirós, Julio D. Rossi. Non-simultaneous blow-up for a quasilinear parabolic system with reaction at the boundary. Communications on Pure & Applied Analysis, 2005, 4 (3) : 523-536. doi: 10.3934/cpaa.2005.4.523
[19]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[20]

Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences