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April  2019, 39(4): 2233-2253. doi: 10.3934/dcds.2019094

Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $

Shengbing Deng ,

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

* Corresponding author: Shengbing Deng

Received  July 2018 Published  January 2019

Fund Project: The author was partially supported financially by NSFC 11501469 and Fundamental Research Funds for the Central Universities XDJK2017B014.

In this paper, we are interested in the following boundary value problem
$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\Delta u +u = \lambda |x-q_1|^{2\alpha_1}\cdots |x-q_n|^{2\alpha_n} u^{p-1}e^{u^p},\ \ u>0,\ \ \ & {\rm in}\ \Omega;\\ \frac{\partial u}{\partial\nu} = 0\ \ \ & {\rm on}\ \partial\Omega, \end{array} \right. \end{eqnarray*} $
where
$ \Omega $
 is a bounded domain in
$ \mathbb{R}^2 $
 with smooth boundary, points
$ q_1,\ldots,q_n\in \Omega $
,
$ \alpha_1,\cdots,\alpha_n\in(0,\infty)\backslash\mathbb{N} $
,
$ \lambda>0 $
 is a small parameter,
$ 0< p <2 $
, and
$ \nu $
 denotes the outer normal vector to
$ \partial\Omega $
. We construct solutions of this problem with
$ k $
 interior bubbling points and
$ l $
 boundary bubbling points, for any
$ k\geq 1 $
 and
$ l\geq 1 $
.
Keywords: Neumann problem, Hénon term, exponential nonlinearity, interior and boundary bubbling solutions, Lyapunov-Schmidt reduction procedure.
Mathematics Subject Classification: Primary: 35J25; Secondary: 35J08, 35J60.
Citation: Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094
References:
[1]

S. Baraket and F. Pacard, 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.  Google Scholar

[2]

D. Bartolucci and G. Tarantello, Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Comm. Math. Phys., 229 (2002), 3-47.  doi: 10.1007/s002200200664.  Google Scholar

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H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u= V(x) e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.  Google Scholar

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D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142.  doi: 10.1007/s002200000302.  Google Scholar

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T. D'Aprile, Multiple blow-up solutions for the Liouville equation with singular data, Comm. Partial Differential Equations, 38 (2013), 1409-1436.  doi: 10.1080/03605302.2013.799487.  Google Scholar

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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.  Google Scholar

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M. del Pino, M. Musso and B. Ruf, New solutions for Trudinger-Moser critical equations in $\mathbb{R}^2$, Journal of Functional Analysis, 258 (2010), 421-457.  doi: 10.1016/j.jfa.2009.06.018.  Google Scholar

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M. del Pino and J. Wei, Collapsing steady states of the Keller–Segel system, Nonlinearity, 19 (2006), 661-684.  doi: 10.1088/0951-7715/19/3/007.  Google Scholar

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S. Deng, Mixed interior and boundary bubbling solutions for Neumann problem in $\mathbb{R}^2$, Journal of Differential Equations, 253 (2012), 727-763.  doi: 10.1016/j.jde.2012.04.012.  Google Scholar

[10]

S. Deng, D. Garrido and M. Musso, Multiple blow-up solutions for an exponential nonlinearity with potential in $\mathbb{R}^2$, Nonlinear Analysis: Theory, Methods and Applications, 119 (2015), 419-442.  doi: 10.1016/j.na.2014.10.034.  Google Scholar

[11]

S. Deng and M. Musso, Bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$, Journal of Differential Equations, 257 (2014), 2259-2302.  doi: 10.1016/j.jde.2014.05.034.  Google Scholar

[12]

S. Deng and M. Musso, Blow up solutions for a Liouville equation with Hénon term, Nonlinear Analysis: Theory, Methods and Applications, 129 (2015), 320-342.  doi: 10.1016/j.na.2015.09.018.  Google Scholar

[13]

P. Esposito, Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal., 36 (2005), 1310-1345.  doi: 10.1137/S0036141003430548.  Google Scholar

[14]

P. Esposito, M. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differential Equations, 227 (2006), 29-68.  doi: 10.1016/j.jde.2006.01.023.  Google Scholar

[15]

P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincar Anal. Non Linaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[16]

Y. Li and I. Shafrir, Blow-up analysis for solutions of $-\Delta u = Ve^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054.  Google Scholar

[17]

L. Ma and J. C. Wei, Convergence for a Liouville equation, Commun. Math. Helv., 76 (2001), 506-514.  doi: 10.1007/PL00013216.  Google Scholar

[18]

K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptot. Anal., 3 (1990), 173-188.   Google Scholar

[19]

T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191-224.   Google Scholar

[20]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal., 191 (2002), 17-51.  doi: 10.1006/jfan.2001.3802.  Google Scholar

[21]

J. Wei, D. Ye and F. Zhou, Bubbling solutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differ. Eq., 28 (2007), 217-247.  doi: 10.1007/s00526-006-0044-y.  Google Scholar

[22]

D. Ye and F. Zhou, A generalized two dimensional Emden-Fowler equation with exponential nonlinearity, Calc. Var. Partial Differ. Eq., 13 (2001), 141-158.  doi: 10.1007/PL00009926.  Google Scholar

[23]

C. Zhao, Singular limits in a Liouville-type equation with singular sources, Houston J. Math., 34 (2008), 601-621.   Google Scholar

show all references

References:
[1]

S. Baraket and F. Pacard, 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.  Google Scholar

[2]

D. Bartolucci and G. Tarantello, Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory, Comm. Math. Phys., 229 (2002), 3-47.  doi: 10.1007/s002200200664.  Google Scholar

[3]

H. Brezis and F. Merle, Uniform estimates and blow-up behavior for solutions of $-\Delta u= V(x) e^u$ in two dimensions, Commun. Partial Differ. Equ., 16 (1991), 1223-1253.  doi: 10.1080/03605309108820797.  Google Scholar

[4]

D. Chae and O. Imanuvilov, The existence of non-topological multivortex solutions in the relativistic self-dual Chern-Simons theory, Comm. Math. Phys., 215 (2000), 119-142.  doi: 10.1007/s002200000302.  Google Scholar

[5]

T. D'Aprile, Multiple blow-up solutions for the Liouville equation with singular data, Comm. Partial Differential Equations, 38 (2013), 1409-1436.  doi: 10.1080/03605302.2013.799487.  Google Scholar

[6]

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.  Google Scholar

[7]

M. del Pino, M. Musso and B. Ruf, New solutions for Trudinger-Moser critical equations in $\mathbb{R}^2$, Journal of Functional Analysis, 258 (2010), 421-457.  doi: 10.1016/j.jfa.2009.06.018.  Google Scholar

[8]

M. del Pino and J. Wei, Collapsing steady states of the Keller–Segel system, Nonlinearity, 19 (2006), 661-684.  doi: 10.1088/0951-7715/19/3/007.  Google Scholar

[9]

S. Deng, Mixed interior and boundary bubbling solutions for Neumann problem in $\mathbb{R}^2$, Journal of Differential Equations, 253 (2012), 727-763.  doi: 10.1016/j.jde.2012.04.012.  Google Scholar

[10]

S. Deng, D. Garrido and M. Musso, Multiple blow-up solutions for an exponential nonlinearity with potential in $\mathbb{R}^2$, Nonlinear Analysis: Theory, Methods and Applications, 119 (2015), 419-442.  doi: 10.1016/j.na.2014.10.034.  Google Scholar

[11]

S. Deng and M. Musso, Bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$, Journal of Differential Equations, 257 (2014), 2259-2302.  doi: 10.1016/j.jde.2014.05.034.  Google Scholar

[12]

S. Deng and M. Musso, Blow up solutions for a Liouville equation with Hénon term, Nonlinear Analysis: Theory, Methods and Applications, 129 (2015), 320-342.  doi: 10.1016/j.na.2015.09.018.  Google Scholar

[13]

P. Esposito, Blow up solutions for a Liouville equation with singular data, SIAM J. Math. Anal., 36 (2005), 1310-1345.  doi: 10.1137/S0036141003430548.  Google Scholar

[14]

P. Esposito, M. Musso and A. Pistoia, Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent, J. Differential Equations, 227 (2006), 29-68.  doi: 10.1016/j.jde.2006.01.023.  Google Scholar

[15]

P. Esposito, M. Grossi and A. Pistoia, On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincar Anal. Non Linaire, 22 (2005), 227-257.  doi: 10.1016/j.anihpc.2004.12.001.  Google Scholar

[16]

Y. Li and I. Shafrir, Blow-up analysis for solutions of $-\Delta u = Ve^u$ in dimension two, Indiana Univ. Math. J., 43 (1994), 1255-1270.  doi: 10.1512/iumj.1994.43.43054.  Google Scholar

[17]

L. Ma and J. C. Wei, Convergence for a Liouville equation, Commun. Math. Helv., 76 (2001), 506-514.  doi: 10.1007/PL00013216.  Google Scholar

[18]

K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptot. Anal., 3 (1990), 173-188.   Google Scholar

[19]

T. Senba and T. Suzuki, Some structures of the solution set for a stationary system of chemotaxis, Adv. Math. Sci. Appl., 10 (2000), 191-224.   Google Scholar

[20]

T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal., 191 (2002), 17-51.  doi: 10.1006/jfan.2001.3802.  Google Scholar

[21]

J. Wei, D. Ye and F. Zhou, Bubbling solutions for an anisotropic Emden-Fowler equation, Calc. Var. Partial Differ. Eq., 28 (2007), 217-247.  doi: 10.1007/s00526-006-0044-y.  Google Scholar

[22]

D. Ye and F. Zhou, A generalized two dimensional Emden-Fowler equation with exponential nonlinearity, Calc. Var. Partial Differ. Eq., 13 (2001), 141-158.  doi: 10.1007/PL00009926.  Google Scholar

[23]

C. Zhao, Singular limits in a Liouville-type equation with singular sources, Houston J. Math., 34 (2008), 601-621.   Google Scholar

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