2012, 11(2): 697-708. doi: 10.3934/cpaa.2012.11.697

On the blow-up boundary solutions of the Monge -Ampére equation with singular weights

1. 

Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Received  July 2010 Revised  July 2011 Published  October 2011

We consider the Monge-Ampére equations det$D^2 u = K(x) f(u)$ in $\Omega$, with $u|_{\partial\Omega}=+\infty$, where $\Omega$ is a bounded and strictly convex smooth domain in $R^N$. When $f(u) = e^u$ or $f(u)= u^p$, $p>N$, and the weight $K(x)\in C^\infty (\Omega )$ grows like a negative power of $d(x)=dist(x, \partial \Omega)$ near $\partial \Omega$, we show some results on the uniqueness, nonexistence and exact boundary blow-up rate of strictly convex solutions for this problem. Existence of such solutions will be also studied in a more general case.
Citation: Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure & Applied Analysis, 2012, 11 (2) : 697-708. doi: 10.3934/cpaa.2012.11.697
References:
[1]

Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior,, J. Anal. Math., 58 (1992), 9.

[2]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampére equation,, Comm. Pure Appl. Math., 37 (1984), 369.

[3]

S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampére equation $det(\partial^2/\partial x_i\partial x_j) =F(x, u)$,, Comm. Pure Appl. Math., 30 (1977), 41.

[4]

S. Y. Cheng and S. T. Yau, On the existence of a complete Kahler metric on non-compact complex manifolds and regularity of Fefferman's equation,, Comm. Pure Appl. Math., 33 (1980), 507.

[5]

M. Chuaqui and C. Cortazar et al., Uniqueness and boundary behavior of large solutions to elliptic problems with weight,, Comm. on Pure and Applied Analysis, 3 (2004), 653.

[6]

F. C. Cirstea and Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations,, Proc. Lond. Math. Soc., 91 (2005), 459.

[7]

F. C. Cirstea and V. Radulescu, Blow-up boundary solutions of semilinear elliptic problems,, Nonlinear Analysis, 48 (2002), 521.

[8]

F. C. Cirstea and C. Trombetti, On the Monge-Ampére equation with boundary blow-up: existence, uniqueness and asymptotics,, Calc. Var. Partial Differential Equations, 31 (2008), 167.

[9]

J. García-Melián and R. Letelier-Albornoz et al., Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up,, Proc. Amer. Math. Soc., 129 (2001), 3593.

[10]

M. Ghergu and V. Radulescu, Nonradial blow-up solutions of sublinear elliptic equations with gradient term,, Comm. on Pure and Applied Analysis, 3 (2004), 465.

[11]

B. Guan and H. Y. Jian, On the Monge-Ampére equation with infinite boundary value,, Pac. J. Math., 216 (2004), 77.

[12]

Y. Huang, Boundary asymptotical behavior of large solutions to Hessian equations,, Pacific J. Math., 244 (2010), 85.

[13]

H. Y. Jian, Hessian equations with infinite Dirichlet boundary,, Indiana Univ. Math. J., 55 (2006), 1045.

[14]

J. B. Keller, On solutions of $\Delta u =f(u)$,, Comm. Pure Appl. Math., 10 (1995), 503.

[15]

N. D. Kutev, Nontrivial solutions for the equations of Monge-Ampére type,, J. Math. Anal. Appl., 132 (1988), 424.

[16]

A. C. Lazer and P. J. Mckenna, On Singular Boundary Value Problems for the Monge-Ampére Operator,, J. Math. Anal. Appl., 197 (1996), 341.

[17]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions,, J. Diff. Eqns, 224 (2006), 385.

[18]

J. Matero, The Bieberbach-Rademacher problem for the Monge-Ampére Operator,, Manuscripta Math., 91 (1996), 379.

[19]

A. Mohammed, On the existence of solutions to the Monge-Ampére equation with infinite boundary values,, Proc. Amer. Math. Soc., 135 (2007), 141.

[20]

A. Mohammed, Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampére equation,, J. Math. Anal. Appl., 340 (2008), 1226.

[21]

H. T. Yang, Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations,, J. Math. Anal. Appl., 314 (2006), 85.

[22]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms and singular weights,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1403.

show all references

References:
[1]

Bandle and M. Marcus, Large solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behavior,, J. Anal. Math., 58 (1992), 9.

[2]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I. Monge-Ampére equation,, Comm. Pure Appl. Math., 37 (1984), 369.

[3]

S. Y. Cheng and S. T. Yau, On the regularity of the Monge-Ampére equation $det(\partial^2/\partial x_i\partial x_j) =F(x, u)$,, Comm. Pure Appl. Math., 30 (1977), 41.

[4]

S. Y. Cheng and S. T. Yau, On the existence of a complete Kahler metric on non-compact complex manifolds and regularity of Fefferman's equation,, Comm. Pure Appl. Math., 33 (1980), 507.

[5]

M. Chuaqui and C. Cortazar et al., Uniqueness and boundary behavior of large solutions to elliptic problems with weight,, Comm. on Pure and Applied Analysis, 3 (2004), 653.

[6]

F. C. Cirstea and Y. Du, General uniqueness results and variation speed for blow-up solutions of elliptic equations,, Proc. Lond. Math. Soc., 91 (2005), 459.

[7]

F. C. Cirstea and V. Radulescu, Blow-up boundary solutions of semilinear elliptic problems,, Nonlinear Analysis, 48 (2002), 521.

[8]

F. C. Cirstea and C. Trombetti, On the Monge-Ampére equation with boundary blow-up: existence, uniqueness and asymptotics,, Calc. Var. Partial Differential Equations, 31 (2008), 167.

[9]

J. García-Melián and R. Letelier-Albornoz et al., Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up,, Proc. Amer. Math. Soc., 129 (2001), 3593.

[10]

M. Ghergu and V. Radulescu, Nonradial blow-up solutions of sublinear elliptic equations with gradient term,, Comm. on Pure and Applied Analysis, 3 (2004), 465.

[11]

B. Guan and H. Y. Jian, On the Monge-Ampére equation with infinite boundary value,, Pac. J. Math., 216 (2004), 77.

[12]

Y. Huang, Boundary asymptotical behavior of large solutions to Hessian equations,, Pacific J. Math., 244 (2010), 85.

[13]

H. Y. Jian, Hessian equations with infinite Dirichlet boundary,, Indiana Univ. Math. J., 55 (2006), 1045.

[14]

J. B. Keller, On solutions of $\Delta u =f(u)$,, Comm. Pure Appl. Math., 10 (1995), 503.

[15]

N. D. Kutev, Nontrivial solutions for the equations of Monge-Ampére type,, J. Math. Anal. Appl., 132 (1988), 424.

[16]

A. C. Lazer and P. J. Mckenna, On Singular Boundary Value Problems for the Monge-Ampére Operator,, J. Math. Anal. Appl., 197 (1996), 341.

[17]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions,, J. Diff. Eqns, 224 (2006), 385.

[18]

J. Matero, The Bieberbach-Rademacher problem for the Monge-Ampére Operator,, Manuscripta Math., 91 (1996), 379.

[19]

A. Mohammed, On the existence of solutions to the Monge-Ampére equation with infinite boundary values,, Proc. Amer. Math. Soc., 135 (2007), 141.

[20]

A. Mohammed, Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampére equation,, J. Math. Anal. Appl., 340 (2008), 1226.

[21]

H. T. Yang, Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations,, J. Math. Anal. Appl., 314 (2006), 85.

[22]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms and singular weights,, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1403.

[1]

Qi-Rui Li, Xu-Jia Wang. Regularity of the homogeneous Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6069-6084. doi: 10.3934/dcds.2015.35.6069

[2]

Alessio Figalli, Young-Heon Kim. Partial regularity of Brenier solutions of the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (2) : 559-565. doi: 10.3934/dcds.2010.28.559

[3]

Bo Guan, Qun Li. A Monge-Ampère type fully nonlinear equation on Hermitian manifolds. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1991-1999. doi: 10.3934/dcdsb.2012.17.1991

[4]

Shouchuan Hu, Haiyan Wang. Convex solutions of boundary value problem arising from Monge-Ampère equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 705-720. doi: 10.3934/dcds.2006.16.705

[5]

Adam M. Oberman. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 221-238. doi: 10.3934/dcdsb.2008.10.221

[6]

Diego Maldonado. On interior \begin{document} $C^2$ \end{document}-estimates for the Monge-Ampère equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1427-1440. doi: 10.3934/dcds.2018058

[7]

Jiakun Liu, Neil S. Trudinger. On Pogorelov estimates for Monge-Ampère type equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1121-1135. doi: 10.3934/dcds.2010.28.1121

[8]

Barbara Brandolini, Carlo Nitsch, Cristina Trombetti. Shape optimization for Monge-Ampère equations via domain derivative. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 825-831. doi: 10.3934/dcdss.2011.4.825

[9]

Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71

[10]

Limei Dai. Multi-valued solutions to a class of parabolic Monge-Ampère equations. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1061-1074. doi: 10.3934/cpaa.2014.13.1061

[11]

Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59

[12]

Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347

[13]

Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447

[14]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[15]

Jorge García-Melián, Julio D. Rossi, José C. Sabina de Lis. Elliptic systems with boundary blow-up: existence, uniqueness and applications to removability of singularities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 549-562. doi: 10.3934/cpaa.2016.15.549

[16]

Zhijun Zhang. Boundary blow-up for elliptic problems involving exponential nonlinearities with nonlinear gradient terms and singular weights. Communications on Pure & Applied Analysis, 2007, 6 (2) : 521-529. doi: 10.3934/cpaa.2007.6.521

[17]

Petri Juutinen. Convexity of solutions to boundary blow-up problems. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2267-2275. doi: 10.3934/cpaa.2013.12.2267

[18]

Shota Sato. Blow-up at space infinity of a solution with a moving singularity for a semilinear parabolic equation. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1225-1237. doi: 10.3934/cpaa.2011.10.1225

[19]

Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809

[20]

Jian Zhang, Shihui Zhu, Xiaoguang Li. Rate of $L^2$-concentration of the blow-up solution for critical nonlinear Schrödinger equation with potential. Mathematical Control & Related Fields, 2011, 1 (1) : 119-127. doi: 10.3934/mcrf.2011.1.119

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]