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Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China |
$ \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*} $ |
$ \Omega $ |
$ \mathbb{R}^2 $ |
$ q_1,\ldots,q_n\in \Omega $ |
$ \alpha_1,\cdots,\alpha_n\in(0,\infty)\backslash\mathbb{N} $ |
$ \lambda>0 $ |
$ 0< p <2 $ |
$ \nu $ |
$ \partial\Omega $ |
$ k $ |
$ l $ |
$ k\geq 1 $ |
$ l\geq 1 $ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
L. Ma and J. C. Wei,
Convergence for a Liouville equation, Commun. Math. Helv., 76 (2001), 506-514.
doi: 10.1007/PL00013216. |
[18] |
K. Nagasaki and T. Suzuki,
Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptot. Anal., 3 (1990), 173-188.
|
[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.
|
[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. |
[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. |
[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. |
[23] |
C. Zhao,
Singular limits in a Liouville-type equation with singular sources, Houston J. Math., 34 (2008), 601-621.
|
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[17] |
L. Ma and J. C. Wei,
Convergence for a Liouville equation, Commun. Math. Helv., 76 (2001), 506-514.
doi: 10.1007/PL00013216. |
[18] |
K. Nagasaki and T. Suzuki,
Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities, Asymptot. Anal., 3 (1990), 173-188.
|
[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.
|
[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. |
[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. |
[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. |
[23] |
C. Zhao,
Singular limits in a Liouville-type equation with singular sources, Houston J. Math., 34 (2008), 601-621.
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