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March  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

Haitao Yang 1, and  Yibin Chang 1,

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.
Keywords: Monge-Ampére equation, uniqueness, singular weight, blow-up solution, boundary behavior..
Mathematics Subject Classification: Primary: 35j05, 35j25; Secondary: 35B4.
Citation: Haitao Yang, Yibin Chang. On the blow-up boundary solutions of the Monge -Ampére equation with singular weights. Communications on Pure and 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), 9-24.

[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), 369-402.

[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), 41-68.

[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), 507-544.

[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), 653-662.

[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), 459-482.

[7]

F. C. Cirstea and V. Radulescu, Blow-up boundary solutions of semilinear elliptic problems, Nonlinear Analysis, T. M. A., 48 (2002), 521-534.

[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), 167-186.

[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), 3593-3602.

[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), 465-474.

[11]

B. Guan and H. Y. Jian, On the Monge-Ampére equation with infinite boundary value, Pac. J. Math., 216 (2004), 77-94.

[12]

Y. Huang, Boundary asymptotical behavior of large solutions to Hessian equations, Pacific J. Math., 244 (2010), 85-98.

[13]

H. Y. Jian, Hessian equations with infinite Dirichlet boundary, Indiana Univ. Math. J., 55 (2006), 1045-1062.

[14]

J. B. Keller, On solutions of $\Delta u =f(u)$, Comm. Pure Appl. Math., 10 (1995), 503-510.

[15]

N. D. Kutev, Nontrivial solutions for the equations of Monge-Ampére type, J. Math. Anal. Appl., 132 (1988), 424-433.

[16]

A. C. Lazer and P. J. Mckenna, On Singular Boundary Value Problems for the Monge-Ampére Operator, J. Math. Anal. Appl., 197 (1996), 341-362.

[17]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Eqns, 224 (2006), 385-439.

[18]

J. Matero, The Bieberbach-Rademacher problem for the Monge-Ampére Operator, Manuscripta Math., 91 (1996), 379-391.

[19]

A. Mohammed, On the existence of solutions to the Monge-Ampére equation with infinite boundary values, Proc. Amer. Math. Soc., 135 (2007), 141-149.

[20]

A. Mohammed, Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampére equation, J. Math. Anal. Appl., 340 (2008), 1226-1234.

[21]

H. T. Yang, Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations, J. Math. Anal. Appl., 314 (2006), 85-96.

[22]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms and singular weights, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1403-1424.

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), 9-24.

[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), 369-402.

[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), 41-68.

[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), 507-544.

[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), 653-662.

[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), 459-482.

[7]

F. C. Cirstea and V. Radulescu, Blow-up boundary solutions of semilinear elliptic problems, Nonlinear Analysis, T. M. A., 48 (2002), 521-534.

[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), 167-186.

[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), 3593-3602.

[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), 465-474.

[11]

B. Guan and H. Y. Jian, On the Monge-Ampére equation with infinite boundary value, Pac. J. Math., 216 (2004), 77-94.

[12]

Y. Huang, Boundary asymptotical behavior of large solutions to Hessian equations, Pacific J. Math., 244 (2010), 85-98.

[13]

H. Y. Jian, Hessian equations with infinite Dirichlet boundary, Indiana Univ. Math. J., 55 (2006), 1045-1062.

[14]

J. B. Keller, On solutions of $\Delta u =f(u)$, Comm. Pure Appl. Math., 10 (1995), 503-510.

[15]

N. D. Kutev, Nontrivial solutions for the equations of Monge-Ampére type, J. Math. Anal. Appl., 132 (1988), 424-433.

[16]

A. C. Lazer and P. J. Mckenna, On Singular Boundary Value Problems for the Monge-Ampére Operator, J. Math. Anal. Appl., 197 (1996), 341-362.

[17]

J. López-Gómez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Eqns, 224 (2006), 385-439.

[18]

J. Matero, The Bieberbach-Rademacher problem for the Monge-Ampére Operator, Manuscripta Math., 91 (1996), 379-391.

[19]

A. Mohammed, On the existence of solutions to the Monge-Ampére equation with infinite boundary values, Proc. Amer. Math. Soc., 135 (2007), 141-149.

[20]

A. Mohammed, Existence and estimates of solutions to a singular Dirichlet problem for the Monge-Ampére equation, J. Math. Anal. Appl., 340 (2008), 1226-1234.

[21]

H. T. Yang, Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations, J. Math. Anal. Appl., 314 (2006), 85-96.

[22]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms and singular weights, Proc. Roy. Soc. Edinburgh Sect. A, 138 (2008), 1403-1424.

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