Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data

Haitao Yang; Yibin Zhang

doi:10.3934/dcds.2017238

Article Contents

2017, Volume 37, Issue 10: 5467-5502. Doi: 10.3934/dcds.2017238

This issue Previous Article Constructing attracting cycles for Halley and Schröder maps of polynomials Next Article

Boundary bubbling solutions for a planar elliptic problem with exponential Neumann data

Haitao Yang^1, and
Yibin Zhang^2, ,

1.
School of Mathematics, Zhejiang University, Hangzhou 310027, China
2.
College of Sciences, Nanjing Agricultural University, Nanjing 210095, China

* Corresponding author: Yibin Zhang
* Corresponding author: Yibin Zhang

Received: February 28, 2017

Revised: April 30, 2017

Published: October 2017

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Abstract

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 $s\to+∞$ .

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Mathematics Subject Classification: Primary: 35B25, 35J65; Secondary: 35B40.

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References

	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.
	I. Babuška and J. Osborn , Eigenvalue problems, Handbook of Numerical Analysis, North-Holland, Amsterdam, 2 (1991) , 641-787.
	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.
	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.
	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.
	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.
	E. N. Dancer and S. Santra , On the superlinear Lazer-McKenna conjecture: The non-homogeneous case, Adv. Differential Equations, 12 (2007) , 961-993.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	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.
	L. Ma and J. Wei , Convergence for a Liouville equation, Comment. Math. Helv., 76 (2001) , 506-514. doi: 10.1007/PL00013216.
	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.
	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.
	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.
	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.
	K. Nagasaki and T. Suzuki , Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptotic Anal., 3 (1990) , 173-188.
	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.
	Y. Wang and L. Wei , Multiple boundary bubbling phenomenon of solutions to a Neumann problem, Adv. Differential Equations, 13 (2008) , 829-856.
	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.
	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.
	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.
	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.
	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.