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

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), 9-24. Google Scholar

[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. Google Scholar

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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. Google Scholar

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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. Google Scholar

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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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[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. Google Scholar

[17]

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

[18]

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

[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. Google Scholar

[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. Google Scholar

[21]

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

[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. Google Scholar

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