- Previous Article
- DCDS Home
- This Issue
-
Next Article
Constructing attracting cycles for Halley and Schröder maps of polynomials
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 |
$\mathbb{R}^2 $ |
$\left\{ \begin{gathered} \begin{gathered} - \Delta \upsilon + \upsilon = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\text{in}}\;\;\Omega {\text{,}} \hfill \\ \frac{{\partial \upsilon }}{{\partial \nu }} = {e^\upsilon } - s{\phi _1} - h\left( x \right)\;\;\;{\text{on}}\;\partial \Omega \hfill \\ \end{gathered} \end{gathered} \right.$ |
$ν$ |
$\partial \Omega$ |
$h∈ C^{0,α}(\partial \Omega)$ |
$s>0$ |
$\phi_1$ |
$\phi_1$ |
$s\to+∞$ |
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. |
[1] |
Heinz Schättler, Urszula Ledzewicz. Lyapunov-Schmidt reduction for optimal control problems. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2201-2223. doi: 10.3934/dcdsb.2012.17.2201 |
[2] |
Christian Pötzsche. Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 739-776. doi: 10.3934/dcdsb.2010.14.739 |
[3] |
Shengbing Deng, Fethi Mahmoudi, Monica Musso. Bubbling on boundary submanifolds for a semilinear Neumann problem near high critical exponents. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3035-3076. doi: 10.3934/dcds.2016.36.3035 |
[4] |
Jean-François Coulombel, Frédéric Lagoutière. The Neumann numerical boundary condition for transport equations. Kinetic and Related Models, 2020, 13 (1) : 1-32. doi: 10.3934/krm.2020001 |
[5] |
Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control and Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015 |
[6] |
Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101 |
[7] |
Zhenghuan Gao, Peihe Wang. Global $ C^2 $-estimates for smooth solutions to uniformly parabolic equations with Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1201-1223. doi: 10.3934/dcds.2021152 |
[8] |
Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623 |
[9] |
Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190 |
[10] |
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 763-800. doi: 10.3934/dcds.2008.21.763 |
[11] |
R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 497-506. doi: 10.3934/dcds.1998.4.497 |
[12] |
Jesús Ildefonso Díaz, L. Tello. On a climate model with a dynamic nonlinear diffusive boundary condition. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 253-262. doi: 10.3934/dcdss.2008.1.253 |
[13] |
Umberto De Maio, Akamabadath K. Nandakumaran, Carmen Perugia. Exact internal controllability for the wave equation in a domain with oscillating boundary with Neumann boundary condition. Evolution Equations and Control Theory, 2015, 4 (3) : 325-346. doi: 10.3934/eect.2015.4.325 |
[14] |
Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 |
[15] |
Tsung-Fang Wu. Multiplicity of positive solutions for a semilinear elliptic equation in $R_+^N$ with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1675-1696. doi: 10.3934/cpaa.2010.9.1675 |
[16] |
M. Eller. On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 473-481. doi: 10.3934/dcdss.2009.2.473 |
[17] |
Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615 |
[18] |
Antonio Cañada, Salvador Villegas. Optimal Lyapunov inequalities for disfocality and Neumann boundary conditions using $L^p$ norms. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 877-888. doi: 10.3934/dcds.2008.20.877 |
[19] |
Shu Luan. On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions. Mathematical Control and Related Fields, 2017, 7 (3) : 493-506. doi: 10.3934/mcrf.2017018 |
[20] |
Shouming Zhou, Chunlai Mu, Yongsheng Mi, Fuchen Zhang. Blow-up for a non-local diffusion equation with exponential reaction term and Neumann boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2935-2946. doi: 10.3934/cpaa.2013.12.2935 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]