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

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

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

Haitao Yang ^1, and Yibin Zhang ^2,,

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

$s\to+∞$

.

Keywords: Nonlinear Neumann boundary condition, boundary bubbling solutions, Lyapunov-Schmidt reduction procedure.

Mathematics Subject Classification: Primary: 35B25, 35J65; Secondary: 35B40.

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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