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

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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

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

Abstract
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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

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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

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