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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

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.
Citation: Liping Wang, Chunyi Zhao. Solutions with clustered bubbles and a boundary layer of an elliptic problem. Discrete and Continuous Dynamical Systems, 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), 9-25. [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), 318-350. doi: 10.1006/jfan.1993.1053. [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), 1324-1356. doi: 10.1016/j.jfa.2013.06.016. [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), 271-297. doi: 10.1002/cpa.3160420304. [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), 19-41. doi: 10.2140/pjm.2001.200.19. [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), 45-82. doi: 10.1016/j.anihpc.2004.05.001. [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), 629-644. doi: 10.1016/j.anihpc.2006.04.004. [8] N. Ghoussoub and C. F. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z., 229 (1998), 443-474. doi: 10.1007/PL00004663. [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), 1929-1946. doi: 10.1081/PDE-100107812. [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), 522-538. doi: 10.4153/CJM-2000-024-x. [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), 252-281. doi: 10.1002/cpa.20139. [12] 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. [13] W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary partial differential equations, North-Holland, Amsterdam, I (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6. [14] W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4. [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), 1-52. doi: 10.1016/0022-1236(90)90002-3. [16] O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains, Comm. Contemp. Math., 1 (1999), 405-449. doi: 10.1142/S0219199799000158. [17] O. Rey, The question of interior blow-up points for an elliptic Neumann problem: the critical case, J. Math. Pures Appl., 81 (2002), 655-696. doi: 10.1016/S0021-7824(01)01251-X. [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), 333-351. doi: 10.3934/dcds.2008.21.333. [19] J. C. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, in Handbook of Differential Equations: Ationary Partial Differential Equations, Elservier/North-Holland, Amsterdam, V (2008), 487-585. doi: 10.1016/S1874-5733(08)80013-7. [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), 1003-1029. doi: 10.1017/S0308210500022617. [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), 311-316. doi: 10.1016/j.crma.2006.07.010. [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), 350-378. doi: 10.1016/j.matpur.2007.07.001.

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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), 9-25. [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), 318-350. doi: 10.1006/jfan.1993.1053. [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), 1324-1356. doi: 10.1016/j.jfa.2013.06.016. [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), 271-297. doi: 10.1002/cpa.3160420304. [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), 19-41. doi: 10.2140/pjm.2001.200.19. [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), 45-82. doi: 10.1016/j.anihpc.2004.05.001. [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), 629-644. doi: 10.1016/j.anihpc.2006.04.004. [8] N. Ghoussoub and C. F. Gui, Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent, Math. Z., 229 (1998), 443-474. doi: 10.1007/PL00004663. [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), 1929-1946. doi: 10.1081/PDE-100107812. [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), 522-538. doi: 10.4153/CJM-2000-024-x. [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), 252-281. doi: 10.1002/cpa.20139. [12] 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. [13] W.-M. Ni, Qualitative properties of solutions to elliptic problems, in Stationary partial differential equations, North-Holland, Amsterdam, I (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6. [14] W.-M. Ni and I. Takagi, Locating the peaks of least energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4. [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), 1-52. doi: 10.1016/0022-1236(90)90002-3. [16] O. Rey, An elliptic Neumann problem with critical nonlinearity in three dimensional domains, Comm. Contemp. Math., 1 (1999), 405-449. doi: 10.1142/S0219199799000158. [17] O. Rey, The question of interior blow-up points for an elliptic Neumann problem: the critical case, J. Math. Pures Appl., 81 (2002), 655-696. doi: 10.1016/S0021-7824(01)01251-X. [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), 333-351. doi: 10.3934/dcds.2008.21.333. [19] J. C. Wei, Existence and stability of spikes for the Gierer-Meinhardt system, in Handbook of Differential Equations: Ationary Partial Differential Equations, Elservier/North-Holland, Amsterdam, V (2008), 487-585. doi: 10.1016/S1874-5733(08)80013-7. [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), 1003-1029. doi: 10.1017/S0308210500022617. [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), 311-316. doi: 10.1016/j.crma.2006.07.010. [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), 350-378. doi: 10.1016/j.matpur.2007.07.001.
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