July  2020, 19(7): 3445-3476. doi: 10.3934/cpaa.2020151

The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data

College of Sciences, Nanjing Agricultural University, Nanjing 210095, China

Received  March 2018 Revised  January 2020 Published  April 2020

Fund Project: This research is supported by the Fundamental Research Funds for the Central Universities under Grant No. KYZ201541 and No. KYZ201649, and the National Natural Science Foundation of China under Grant No. 11601232, No. 11671354 and No. 11775116

Let
$ \Omega\subset\mathbb{R}^2 $
be a bounded smooth domain, we study the following anisotropic elliptic problem
$ \begin{cases} -\nabla(a(x)\nabla \upsilon)+a(x)\upsilon = 0& \text{in}\, \, \, \, \, \Omega, \\ \dfrac{\partial \upsilon}{\partial\nu} = e^\upsilon-s\phi_1-h(x) & \text{on}\, \ \, \partial\Omega, \end{cases} $
where
$ \nu $
denotes the outer unit normal vector to
$ \partial\Omega $
,
$ h\in C^{0, \alpha}( \partial\Omega) $
,
$ s>0 $
is a large parameter,
$ a(x) $
is a positive smooth function and
$ \phi_1 $
is a positive first Steklov eigenfunction. We show that this problem has an unbounded number of solutions for all sufficiently large
$ s $
, which give a positive answer to a generalization of the Lazer-McKenna conjecture for this case. Moreover, the solutions found exhibit multiple concentration behavior around boundary maxima of
$ a(x)\phi_1 $
as
$ s\rightarrow+\infty $
.
Citation: Yibin Zhang. The Lazer-McKenna conjecture for an anisotropic planar elliptic problem with exponential Neumann data. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3445-3476. doi: 10.3934/cpaa.2020151
References:
[1]

A. Ambrosetti and G. Prodi, On the inversion of some differentiable mappings with singularities between Banach spaces, Ann. Mat. Pura Appl., 93 (1973), 231-247.  doi: 10.1007/BF02412022.

[2]

I. Babuška and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, Vol. Ⅱ, North-Holland, Amsterdam, (1991), 641–787.

[3]

B. BreuerP. J. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof, J. Differ. Equ., 195 (2003), 243-269.  doi: 10.1016/S0022-0396(03)00186-4.

[4]

E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: the non-homogeneous case, Adv. Differ. Equ., 12 (2007), 961-993. 

[5]

E. N. Dancer and S. Yan, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Meth. Appl. Anal., 15, (2008), 97–119. doi: 10.4310/MAA.2008.v15.n1.a9.

[6]

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.

[7]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part II, Commun. Partial Differ. Equ., 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865.

[8]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differ. Equ., 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017.

[9]

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.

[10]

M. del Pino and C. Muñoz, The two-dimensional Lazer-Mckenna conjecture for an exponential nonlinearity, J. Differ. Equ., 231 (2006), 108-134.  doi: 10.1016/j.jde.2006.07.003.

[11]

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.

[12]

O. Druet, The critical Lazer-McKenna conjecture in low dimensions, J. Differ. Equ., 245 (2008), 2199-2242.  doi: 10.1016/j.jde.2008.05.002.

[13]

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.

[14]

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.

[15]

G. LiS. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial Differ. Equ., 28 (2007), 471-508.  doi: 10.1007/s00526-006-0051-z.

[16]

G. LiS. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, part II, J. Differ. Equ., 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011.

[17]

R. Molle and D. Passaseo, Elliptic equations with jumping nonlinearities involving high eigenvalues, Calc. Var. Partial Differ. Equ., 49 (2014), 861-907.  doi: 10.1007/s00526-013-0603-y.

[18]

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.

[19]

R. Molle and D. Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities, Ann. Inst. Henri Poincare Anal. Non Lineaire, 27 (2010), 529-553.  doi: 10.1016/j.anihpc.2009.09.005.

[20]

B. Ou, A uniqueness theorem for harmonic functions on the upper-half plane, Conform. Geom. Dyn., 4 (2000), 120-125.  doi: 10.1090/S1088-4173-00-00067-9.

[21]

Y. Wang and L. Wei, Multiple boundary bubbling phenomenon of solutions to a Neumann problem, Adv. Differ. Equ., 13 (2008), 829-856. 

[22]

J. Wei and S. Yan, On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 9 (2010), 423-457. 

[23]

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.

[24]

H. Yang and Y. Zhang, Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data, Discrete Contin. Dyn. Syst. A, 37 (2017), 5467-5502.  doi: 10.3934/dcds.2017238.

[25]

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 Differ. Equ., 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 (1973), 231-247.  doi: 10.1007/BF02412022.

[2]

I. Babuška and J. Osborn, Eigenvalue problems, in Handbook of Numerical Analysis, Vol. Ⅱ, North-Holland, Amsterdam, (1991), 641–787.

[3]

B. BreuerP. J. McKenna and M. Plum, Multiple solutions for a semilinear boundary value problem: A computational multiplicity proof, J. Differ. Equ., 195 (2003), 243-269.  doi: 10.1016/S0022-0396(03)00186-4.

[4]

E. N. Dancer and S. Santra, On the superlinear Lazer-McKenna conjecture: the non-homogeneous case, Adv. Differ. Equ., 12 (2007), 961-993. 

[5]

E. N. Dancer and S. Yan, On the Lazer-McKenna conjecture involving critical and supercritical exponents, Meth. Appl. Anal., 15, (2008), 97–119. doi: 10.4310/MAA.2008.v15.n1.a9.

[6]

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.

[7]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, part II, Commun. Partial Differ. Equ., 30 (2005), 1331-1358.  doi: 10.1080/03605300500258865.

[8]

E. N. Dancer and S. Yan, On the superlinear Lazer-McKenna conjecture, J. Differ. Equ., 210 (2005), 317-351.  doi: 10.1016/j.jde.2004.07.017.

[9]

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.

[10]

M. del Pino and C. Muñoz, The two-dimensional Lazer-Mckenna conjecture for an exponential nonlinearity, J. Differ. Equ., 231 (2006), 108-134.  doi: 10.1016/j.jde.2006.07.003.

[11]

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.

[12]

O. Druet, The critical Lazer-McKenna conjecture in low dimensions, J. Differ. Equ., 245 (2008), 2199-2242.  doi: 10.1016/j.jde.2008.05.002.

[13]

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.

[14]

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.

[15]

G. LiS. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, Calc. Var. Partial Differ. Equ., 28 (2007), 471-508.  doi: 10.1007/s00526-006-0051-z.

[16]

G. LiS. Yan and J. Yang, The Lazer-McKenna conjecture for an elliptic problem with critical growth, part II, J. Differ. Equ., 227 (2006), 301-332.  doi: 10.1016/j.jde.2006.02.011.

[17]

R. Molle and D. Passaseo, Elliptic equations with jumping nonlinearities involving high eigenvalues, Calc. Var. Partial Differ. Equ., 49 (2014), 861-907.  doi: 10.1007/s00526-013-0603-y.

[18]

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.

[19]

R. Molle and D. Passaseo, Multiple solutions for a class of elliptic equations with jumping nonlinearities, Ann. Inst. Henri Poincare Anal. Non Lineaire, 27 (2010), 529-553.  doi: 10.1016/j.anihpc.2009.09.005.

[20]

B. Ou, A uniqueness theorem for harmonic functions on the upper-half plane, Conform. Geom. Dyn., 4 (2000), 120-125.  doi: 10.1090/S1088-4173-00-00067-9.

[21]

Y. Wang and L. Wei, Multiple boundary bubbling phenomenon of solutions to a Neumann problem, Adv. Differ. Equ., 13 (2008), 829-856. 

[22]

J. Wei and S. Yan, On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 9 (2010), 423-457. 

[23]

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.

[24]

H. Yang and Y. Zhang, Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data, Discrete Contin. Dyn. Syst. A, 37 (2017), 5467-5502.  doi: 10.3934/dcds.2017238.

[25]

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 Differ. Equ., 16 (2003), 405-430.  doi: 10.1007/s005260100155.

[1]

Yuxia Guo, Ting Liu. Lazer-McKenna conjecture for higher order elliptic problem with critical growth. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 1159-1189. doi: 10.3934/dcds.2020074

[2]

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

[3]

Lisa Hollman, P. J. McKenna. A conjecture on multiple solutions of a nonlinear elliptic boundary value problem: some numerical evidence. Communications on Pure and Applied Analysis, 2011, 10 (2) : 785-802. doi: 10.3934/cpaa.2011.10.785

[4]

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

[5]

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

[6]

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

[7]

Bhakti Bhusan Manna, Sanjiban Santra. On the Hollman McKenna conjecture: Interior concentration near curves. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5595-5626. doi: 10.3934/dcds.2016046

[8]

Patrick Winkert. Multiplicity results for a class of elliptic problems with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2013, 12 (2) : 785-802. doi: 10.3934/cpaa.2013.12.785

[9]

J. García-Melián, Julio D. Rossi, José Sabina de Lis. A convex-concave elliptic problem with a parameter on the boundary condition. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1095-1124. doi: 10.3934/dcds.2012.32.1095

[10]

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

[11]

G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 279-294. doi: 10.3934/dcdsb.2002.2.279

[12]

Grégoire Allaire, Yves Capdeboscq, Marjolaine Puel. Homogenization of a one-dimensional spectral problem for a singularly perturbed elliptic operator with Neumann boundary conditions. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 1-31. doi: 10.3934/dcdsb.2012.17.1

[13]

Jiayue Zheng, Shangbin Cui. Bifurcation analysis of a tumor-model free boundary problem with a nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4397-4410. doi: 10.3934/dcdsb.2020103

[14]

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

[15]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340

[16]

Costică Moroşanu, Bianca Satco. Qualitative and quantitative analysis for a nonlocal and nonlinear reaction-diffusion problem with in-homogeneous Neumann boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022042

[17]

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

[18]

Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1023-1048. doi: 10.3934/cpaa.2019050

[19]

Liping Wang, Dong Ye. Concentrating solutions for an anisotropic elliptic problem with large exponent. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3771-3797. doi: 10.3934/dcds.2015.35.3771

[20]

Everaldo S. de Medeiros, Jianfu Yang. Asymptotic behavior of solutions to a perturbed p-Laplacian problem with Neumann condition. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 595-606. doi: 10.3934/dcds.2005.12.595

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (259)
  • HTML views (76)
  • Cited by (0)

Other articles
by authors

[Back to Top]