The Hénon equation with a critical exponent under the Neumann boundary condition

Jaeyoung Byeon; Sangdon Jin

doi:10.3934/dcds.2018190

Article Contents

2018, Volume 38, Issue 9: 4353-4390. Doi: 10.3934/dcds.2018190

This issue Previous Article Traveling wave solutions for time periodic reaction-diffusion systems Next Article Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)

The Hénon equation with a critical exponent under the Neumann boundary condition

Jaeyoung Byeon and
Sangdon Jin

Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea

Received: July 31, 2017

Revised: March 31, 2018

Published: August 2018

Abstract Full Text(HTML) Related Papers Cited by

Abstract

For $n≥ 3$ and $p = (n+2)/(n-2), $ we consider the Hénon equation with the homogeneous Neumann boundary condition

$ -Δ u + u = |x|^{α}u^{p}, \; u > 0 \;\text{in} \; Ω,\ \ \frac{\partial u}{\partial ν} = 0 \; \text{ on }\;\partial Ω,$

where $Ω \subset B(0,1) \subset \mathbb{R}^n, n ≥ 3$, $α≥ 0$ and $\partial^*Ω \equiv \partialΩ \cap \partial B(0,1) \ne \emptyset.$ It is well known that for $α = 0,$ there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for $α > 0$ and its asymptotic behavior as the parameter $α$ approaches from below to a threshold $α_0 ∈ (0,∞]$ for existence of a least energy solution.

Keywords:

Mathematics Subject Classification: Primary: 35B33, 35B40, 35J15, 35J25, 35J61.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	Adimurthi and G. Mancini , The Neumann problem for elliptic equations with critical nonlinearity, A tribute in honour of G. Prodi, Scu. Norm. Sup. Pisa, (1991) , 9-25.
	Adimurthi , G. Mancini and S. L. Yadava , The role of the mean curvature in semilinear Neumann problem involving critical exponent, Comm. Partial Differential Equations, 20 (1995) , 591-631. doi: 10.1080/03605309508821110.
	Adimurthi , F. Pacella and S. L. Yadava , Characterization of concentration points and $L^∞$ -estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent, Differential Integral Equations, 8 (1995) , 41-68.
	Adimurthi , F. Pacella and S. L. Yadava , Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity, J. Funct. Anal., 113 (1993) , 318-350. doi: 10.1006/jfan.1993.1053.
	M. Badiale and E. Serra , Multiplicity results for the supercritical Hénon equation, Adv. Nonlinear Studies, 4 (2004) , 453-467. doi: 10.1515/ans-2004-0406.
	H. Brézis and L. Nirenberg , Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983) , 437-477. doi: 10.1002/cpa.3160360405.
	J. Byeon , S. Cho and J. Park , On the location of a peak point of a least energy solution for Hénon equation, Discrete Contin. Dyn. Syst., 30 (2011) , 1055-1081. doi: 10.3934/dcds.2011.30.1055.
	J. Byeon and Z.-Q. Wang , On the Hénon equation: Asymptotic profile of ground states, Ann. Inst. H. Poincare Anal. Non Lineaire, 23 (2006) , 803-828. doi: 10.1016/j.anihpc.2006.04.001.
	J. Byeon and Z.-Q. Wang , On the Hénon equation: Asymptotic profile of ground states $\amalg$, J. Differential Equations, 216 (2005) , 78-108. doi: 10.1016/j.jde.2005.02.018.
	J. Byeon and Z. Q. Wang , On the Hénon equation with a Neumann boundary condition: Asymptotic profile of ground states, Journal of Functional Analysis, 274 (2018) , 3325-3376. doi: 10.1016/j.jfa.2018.03.015.
	D. Cao and S. Peng , The asymptotic behaviour of the ground state solutions for Henon equation, J. Math. Anal. Appl., 278 (2003) , 1-17. doi: 10.1016/S0022-247X(02)00292-5.
	D. Cao , S. Peng and S. Yan , Asymptotic behaviour of ground state solutions for the Henon equation, IMA J. Appl. Math., 74 (2009) , 468-480. doi: 10.1093/imamat/hxn035.
	J. Chabrowski and M. Willem , Least energy solutions of a critical Neumann problem with a weight, Calc. Var. Partial Differential Equations, 15 (2002) , 421-431. doi: 10.1007/s00526-002-0101-0.
	G. Chen , W. M. Ni and J. Zhou , Algorithms and visualization for solutions of nonlinear ellptic equations, Inter. Jour. Bifur. Chaos, 10 (2000) , 1565-1612. doi: 10.1142/S0218127400001006.
	D. G. Costa and P. M. Girão , Existence and nonexistence of least energy solutions of the Neumann problem for a semilinear elliptic equation with critical Sobolev exponent and a critical lower-order perturbations, J. Differential Equations, 188 (2003) , 164-202. doi: 10.1016/S0022-0396(02)00070-0.
	M. Gazzini and E. Serra , The Neumann problem for the Henon equation, trace inequalities and Steklov eigenvalues, Ann. Inst. H. Poincare Anal. Non Lineaire, 25 (2008) , 281-302. doi: 10.1016/j.anihpc.2006.09.003.
	D. Gilbarg and N. Trudinger, Elliptic Partial Differntial Equations of Second Order, 2^nd edition, Grundlehren 224, Springer, Berlin, Heidelberg, New York and Tokyo, 1983. doi: 10.1007/978-3-642-61798-0.
	M. Hénon , Numerical experiments on the stability of spherical stellar systems, Astronomy and Astrophysics, 24 (1973) , 229-238.
	C. S. Lin , W. M. Ni and I. Takagi , Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988) , 1-27. doi: 10.1016/0022-0396(88)90147-7.
	P. L. Lions , The concentration-compactness principle in the calculus of variations, The limit case Ⅱ, Rev. Mat. Iberoamericana, 1 (1985) , 45-121. doi: 10.4171/RMI/12.
	P. L. Lions , F. Pacella and M. Tricarico , Best constants in Sobolev inequalities for functions vanishing on some part of the boundary and related questions, Indiana Univ. Math. J., 37 (1988) , 301-324. doi: 10.1512/iumj.1988.37.37015.
	W. M. Ni , A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J., 31 (1982) , 801-807. doi: 10.1512/iumj.1982.31.31056.
	W. M. Ni , X. B. Pan and I. Takagi , Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents, Duke Math. J., 67 (1992) , 1-20. doi: 10.1215/S0012-7094-92-06701-9.
	W. M. Ni and I. Takagi , On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math., 44 (1991) , 819-851. doi: 10.1002/cpa.3160440705.
	S. Secchi and E. Serra , Symmetry breaking results for problems with exponential growth in the unit disc, Comm. Contemp. Math., 8 (2006) , 823-839. doi: 10.1142/S0219199706002295.
	E. Serra , Non radial positive solutions for the Hénon equation with critical growth, Calc. Var. Partial Differential Equations, 23 (2005) , 301-326. doi: 10.1007/s00526-004-0302-9.
	D. Smets , J. Su and M. Willem , Non-radial ground states for the Hénon eqaution, Communications in Contemporary Mathematics, 4 (2002) , 467-480. doi: 10.1142/S0219199702000725.
	D. Smets , Partial symmetry and asymptotic behavior for some elliptic variational problems, Calc. Var. Partial Differential Equations, 18 (2003) , 57-75. doi: 10.1007/s00526-002-0180-y.
	X. J. Wang , Neumann problems of semilinear elliptic equations involving critical Sobolev exponents, J. Differential Equations, 93 (1991) , 283-310. doi: 10.1016/0022-0396(91)90014-Z.
	J. Wei and S. Yan , Infinitely many nonradial solutions for the Hénon equation with critical growth, Rev. Mat. Iberoamericana, 29 (2013) , 997-1020. doi: 10.4171/RMI/747.