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 & 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.  Google Scholar

[2]

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

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[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.  Google Scholar

[4]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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