# American Institute of Mathematical Sciences

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

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

 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
 $l\geq 1$
.
Citation: Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems - A, 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 [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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##### 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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