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 & Continuous Dynamical Systems - A, 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. BreuerP. 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. DávilaM. 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 FigueiredoP. 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 PinoM. 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. EspositoM. 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. LiS. 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. LiS. 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. WeiD. 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. BreuerP. 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. DávilaM. 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 FigueiredoP. 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 PinoM. 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. EspositoM. 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. LiS. 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. LiS. 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. WeiD. 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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