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

Yibin Zhang

doi:10.3934/cpaa.2020151

Article Contents

2020, Volume 19, Issue 7: 3445-3476. Doi: 10.3934/cpaa.2020151

This issue Previous Article Next Article Generalizations of

$ p $

-Laplace operator for image enhancement: Part 2

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

Yibin Zhang

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

Received: March 2018

Revised: January 2020

Published: April 2020

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

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

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 $.

Keywords:

Mathematics Subject Classification: Primary 35B25, 35J25; Secondary 35B40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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. Breuer, P. 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. 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.
[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 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.
[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. Li, S. 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. Li, S. 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.