# American Institute of Mathematical Sciences

October  2017, 37(10): 5467-5502. doi: 10.3934/dcds.2017238

## Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data

 1 School of Mathematics, Zhejiang University, Hangzhou 310027, China 2 College of Sciences, Nanjing Agricultural University, Nanjing 210095, China

* Corresponding author: Yibin Zhang

Received  March 2017 Revised  May 2017 Published  June 2017

Let Ω be a bounded domain in
 $\mathbb{R}^2$
with smooth boundary, we study the following Neumann boundary value problem
 $\left\{ \begin{gathered} \begin{gathered} - \Delta \upsilon + \upsilon = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{in}}\;\;\Omega {\text{,}} \hfill \\ \frac{{\partial \upsilon }}{{\partial \nu }} = {e^\upsilon } - s{\phi _1} - h\left( x \right)\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \end{gathered} \right.$
where
 $ν$
denotes the outer unit normal vector to
 $\partial \Omega$
,
 $h∈ C^{0,α}(\partial \Omega)$
,
 $s>0$
is a large parameter and
 $\phi_1$
is a positive first Steklov eigenfunction. We construct solutions of this problem which exhibit multiple boundary concentration behavior around maximum points of
 $\phi_1$
on the boundary as
 $s\to+∞$
.
Citation: Haitao Yang, Yibin Zhang. Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5467-5502. doi: 10.3934/dcds.2017238
##### References:
 [1] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl., 93 (1972), 231-247.  doi: 10.1007/BF02412022. [2] I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, North-Holland, Amsterdam, 2 (1991), 641-787. [3] S. Baraket and F. Parcard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations, 6 (1998), 1-38.  doi: 10.1007/s005260050080. [4] B. Breuer, P. J. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof, J. Differential Equations, 195 (2003), 243-269.  doi: 10.1016/S0022-0396(03)00186-4. [5] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-Δ u=V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797. [6] C. Chen and C. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727.  doi: 10.1002/cpa.10107. [7] E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The non-homogeneous case, Adv. Differential Equations, 12 (2007), 961-993. [8] E. N. Dancer and S. Yan, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Methods Appl. Anal., 15 (2008), 97-119.  doi: 10.4310/MAA.2008.v15.n1.a9. [9] E. N. Dancer and S. Yan, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc., 78 (2008), 639-662.  doi: 10.1112/jlms/jdn045. [10] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part Ⅱ, Comm. Partial Differential Equations, 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865. [11] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017. [12] J. Dávila, M. del Pino and M. Musso, Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data, J. Funct. Anal., 227 (2005), 430-490.  doi: 10.1016/j.jfa.2005.06.010. [13] D. G. de Figueiredo, P. N. Srikanth and S. Santra, Non-radially symmetric solutions for a superlinear Ambrosetti-Prodi type problem in a ball, Commun. Contemp. Math., 7 (2005), 849-866.  doi: 10.1142/S0219199705001982. [14] M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5. [15] M. del Pino and C. Muñz, The two-dimensional Lazer-Mckenna conjecture for an exponential nonlinearity, J. Differential Equations, 231 (2006), 108-134.  doi: 10.1016/j.jde.2006.07.003. [16] O. Druet, The critical Lazer-McKenna conjecture in low dimensions, J. Differential Equations, 245 (2008), 2199-2242.  doi: 10.1016/j.jde.2008.05.002. [17] P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Analyse Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001. [18] O. Kavian and M. Vogelius, On the existence and "blow-up" of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modelling, Proc. Roy. Soc. Edinburgh Section A, 133 (2003), 119-149.  doi: 10.1017/S0308210500002316. [19] A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294.  doi: 10.1016/0022-247X(81)90166-9. [20] Y. Li and I. Shafrir, Blow-up analysis for solutions of $-Δ u =Ve^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054. [21] Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8. [22] G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial Differential Equations, 28 (2007), 471-508.  doi: 10.1007/s00526-006-0051-z. [23] G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, part Ⅱ, J. Differential Equations, 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011. [24] L. Ma and J. Wei, Convergence for a Liouville equation, Comment. Math. Helv., 76 (2001), 506-514.  doi: 10.1007/PL00013216. [25] K. Medville and M. Vogelius, Blow up behavior of planar harmonic functions satisfying a certain exponential Neumann boundary condition, SIAM J. Math. Anal., 36 (2005), 1772-1806.  doi: 10.1137/S0036141003436090. [26] R. Molle and D. Passaseo, Elliptic equations with jumping nonlinearities involving high eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 861-907.  doi: 10.1007/s00526-013-0603-y. [27] R. Molle and D. Passaseo, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities, J. Funct. Anal., 259 (2010), 2253-2295.  doi: 10.1016/j.jfa.2010.05.010. [28] R. Molle and D. Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 529-553.  doi: 10.1016/j.anihpc.2009.09.005. [29] K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal., 3 (1990), 173-188. [30] B. Ou, A uniqueness theorem for harmonic functions on the upper-half plane, Conform. Geom. Dynamics, 4 (2000), 120-125.  doi: 10.1090/S1088-4173-00-00067-9. [31] Y. Wang and L. Wei, Multiple boundary bubbling phenomenon of solutions to a Neumann problem, Adv. Differential Equations, 13 (2008), 829-856. [32] L. Wei, Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions, Comm. Pure Appl. Anal., 7 (2008), 925-946.  doi: 10.3934/cpaa.2008.7.925. [33] J. Wei and S. Yan, On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 9 (2010), 423-457. [34] J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667.  doi: 10.1016/j.jfa.2006.11.002. [35] J. Wei, D. Ye and F. Zhou, Bubbling solutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differential Equations, 28 (2007), 217-247.  doi: 10.1007/s00526-006-0044-y. [36] L. Zhang, Classification of conformal metrics on $\mathbb{R}^2_+$ with constant Gauss curvature and geodesic curvature on the boundary under various integral finiteness assumptions, Calc. Var. Partial Differential Equations, 16 (2003), 405-430.  doi: 10.1007/s005260100155.

show all references

##### References:
 [1] A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl., 93 (1972), 231-247.  doi: 10.1007/BF02412022. [2] I. Babuška and J. Osborn, Eigenvalue problems, Handbook of Numerical Analysis, North-Holland, Amsterdam, 2 (1991), 641-787. [3] S. Baraket and F. Parcard, Construction of singular limits for a semilinear elliptic equation in dimension 2, Calc. Var. Partial Differential Equations, 6 (1998), 1-38.  doi: 10.1007/s005260050080. [4] B. Breuer, P. J. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof, J. Differential Equations, 195 (2003), 243-269.  doi: 10.1016/S0022-0396(03)00186-4. [5] H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-Δ u=V(x)e^u$ in two dimensions, Comm. Partial Differential Equations, 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797. [6] C. Chen and C. Lin, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure Appl. Math., 56 (2003), 1667-1727.  doi: 10.1002/cpa.10107. [7] E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: The non-homogeneous case, Adv. Differential Equations, 12 (2007), 961-993. [8] E. N. Dancer and S. Yan, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Methods Appl. Anal., 15 (2008), 97-119.  doi: 10.4310/MAA.2008.v15.n1.a9. [9] E. N. Dancer and S. Yan, The Lazer-McKenna conjecture and a free boundary problem in two dimensions, J. Lond. Math. Soc., 78 (2008), 639-662.  doi: 10.1112/jlms/jdn045. [10] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part Ⅱ, Comm. Partial Differential Equations, 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865. [11] E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differential Equations, 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017. [12] J. Dávila, M. del Pino and M. Musso, Concentrating solutions in a two-dimensional elliptic problem with exponential Neumann data, J. Funct. Anal., 227 (2005), 430-490.  doi: 10.1016/j.jfa.2005.06.010. [13] D. G. de Figueiredo, P. N. Srikanth and S. Santra, Non-radially symmetric solutions for a superlinear Ambrosetti-Prodi type problem in a ball, Commun. Contemp. Math., 7 (2005), 849-866.  doi: 10.1142/S0219199705001982. [14] M. del Pino, M. Kowalczyk and M. Musso, Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81.  doi: 10.1007/s00526-004-0314-5. [15] M. del Pino and C. Muñz, The two-dimensional Lazer-Mckenna conjecture for an exponential nonlinearity, J. Differential Equations, 231 (2006), 108-134.  doi: 10.1016/j.jde.2006.07.003. [16] O. Druet, The critical Lazer-McKenna conjecture in low dimensions, J. Differential Equations, 245 (2008), 2199-2242.  doi: 10.1016/j.jde.2008.05.002. [17] P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Analyse Non Linéaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001. [18] O. Kavian and M. Vogelius, On the existence and "blow-up" of solutions to a two-dimensional nonlinear boundary-value problem arising in corrosion modelling, Proc. Roy. Soc. Edinburgh Section A, 133 (2003), 119-149.  doi: 10.1017/S0308210500002316. [19] A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl., 84 (1981), 282-294.  doi: 10.1016/0022-247X(81)90166-9. [20] Y. Li and I. Shafrir, Blow-up analysis for solutions of $-Δ u =Ve^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054. [21] Y. Li and M. Zhu, Uniqueness theorems through the method of moving spheres, Duke Math. J., 80 (1995), 383-417.  doi: 10.1215/S0012-7094-95-08016-8. [22] G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial Differential Equations, 28 (2007), 471-508.  doi: 10.1007/s00526-006-0051-z. [23] G. Li, S. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, part Ⅱ, J. Differential Equations, 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011. [24] L. Ma and J. Wei, Convergence for a Liouville equation, Comment. Math. Helv., 76 (2001), 506-514.  doi: 10.1007/PL00013216. [25] K. Medville and M. Vogelius, Blow up behavior of planar harmonic functions satisfying a certain exponential Neumann boundary condition, SIAM J. Math. Anal., 36 (2005), 1772-1806.  doi: 10.1137/S0036141003436090. [26] R. Molle and D. Passaseo, Elliptic equations with jumping nonlinearities involving high eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 861-907.  doi: 10.1007/s00526-013-0603-y. [27] R. Molle and D. Passaseo, Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities, J. Funct. Anal., 259 (2010), 2253-2295.  doi: 10.1016/j.jfa.2010.05.010. [28] R. Molle and D. Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 529-553.  doi: 10.1016/j.anihpc.2009.09.005. [29] K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal., 3 (1990), 173-188. [30] B. Ou, A uniqueness theorem for harmonic functions on the upper-half plane, Conform. Geom. Dynamics, 4 (2000), 120-125.  doi: 10.1090/S1088-4173-00-00067-9. [31] Y. Wang and L. Wei, Multiple boundary bubbling phenomenon of solutions to a Neumann problem, Adv. Differential Equations, 13 (2008), 829-856. [32] L. Wei, Concentrating phenomena in some elliptic Neumann problem: Asymptotic behavior of solutions, Comm. Pure Appl. Anal., 7 (2008), 925-946.  doi: 10.3934/cpaa.2008.7.925. [33] J. Wei and S. Yan, On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 9 (2010), 423-457. [34] J. Wei and S. Yan, Lazer-McKenna conjecture: The critical case, J. Funct. Anal., 244 (2007), 639-667.  doi: 10.1016/j.jfa.2006.11.002. [35] J. Wei, D. Ye and F. Zhou, Bubbling solutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differential Equations, 28 (2007), 217-247.  doi: 10.1007/s00526-006-0044-y. [36] L. Zhang, Classification of conformal metrics on $\mathbb{R}^2_+$ with constant Gauss curvature and geodesic curvature on the boundary under various integral finiteness assumptions, Calc. Var. Partial Differential Equations, 16 (2003), 405-430.  doi: 10.1007/s005260100155.
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