On the existence of solutions to the Monge-Ampère equation with infinite boundary values
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Abstract:
Given a positive and an increasing nonlinearity $f$ that satisfies an appropriate growth condition at infinity, we provide a condition on $g\in C^\infty (\Omega )$ for which the Monge-Ampère equation $\operatorname {det} D^2u=gf(u)$ admits a solution with infinite boundary value on a strictly convex domain $\Omega$. Sufficient conditions for the nonexistence of such solutions will also be given.References
- Catherine Bandle and Moshe Marcus, “Large” solutions of semilinear elliptic equations: existence, uniqueness and asymptotic behaviour, J. Anal. Math. 58 (1992), 9–24. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. MR 1226934, DOI 10.1007/BF02790355
- Catherine Bandle and Moshe Marcus, Asymptotic behaviour of solutions and their derivatives, for semilinear elliptic problems with blowup on the boundary, Ann. Inst. H. Poincaré C Anal. Non Linéaire 12 (1995), no. 2, 155–171. MR 1326666, DOI 10.1016/S0294-1449(16)30162-7
- Ludwig Bieberbach, $\Delta u=e^u$ und die automorphen Funktionen, Math. Ann. 77 (1916), no. 2, 173–212 (German). MR 1511854, DOI 10.1007/BF01456901
- L. Caffarelli, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equation, Comm. Pure Appl. Math. 37 (1984), no. 3, 369–402. MR 739925, DOI 10.1002/cpa.3160370306
- Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the Monge-Ampère equation $\textrm {det}(\partial ^{2}u/\partial x_{i}\partial sx_{j})=F(x,u)$, Comm. Pure Appl. Math. 30 (1977), no. 1, 41–68. MR 437805, DOI 10.1002/cpa.3160300104
- Shiu Yuen Cheng and Shing Tung Yau, On the existence of a complete Kähler metric on noncompact complex manifolds and the regularity of Fefferman’s equation, Comm. Pure Appl. Math. 33 (1980), no. 4, 507–544. MR 575736, DOI 10.1002/cpa.3160330404
- Shiu Yuen Cheng and Shing-Tung Yau, The real Monge-Ampère equation and affine flat structures, Proceedings of the 1980 Beijing Symposium on Differential Geometry and Differential Equations, Vol. 1, 2, 3 (Beijing, 1980) Sci. Press Beijing, Beijing, 1982, pp. 339–370. MR 714338
- Florica-Corina Şt. Cîrstea and Vicenţiu D. Rădulescu, Existence and uniqueness of blow-up solutions for a class of logistic equations, Commun. Contemp. Math. 4 (2002), no. 3, 559–586. MR 1918760, DOI 10.1142/S0219199702000737
- G. Díaz and R. Letelier, Explosive solutions of quasilinear elliptic equations: existence and uniqueness, Nonlinear Anal. 20 (1993), no. 2, 97–125. MR 1200384, DOI 10.1016/0362-546X(93)90012-H
- Yihong Du and Zongming Guo, Boundary blow-up solutions and their applications in quasilinear elliptic equations, J. Anal. Math. 89 (2003), 277–302. MR 1981921, DOI 10.1007/BF02893084
- J. García-Melián, R. Letelier-Albornoz, and J. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc. 129 (2001), no. 12, 3593–3602. MR 1860492, DOI 10.1090/S0002-9939-01-06229-3
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364
- Francesca Gladiali and Giovanni Porru, Estimates for explosive solutions to $p$-Laplace equations, Progress in partial differential equations, Vol. 1 (Pont-à-Mousson, 1997) Pitman Res. Notes Math. Ser., vol. 383, Longman, Harlow, 1998, pp. 117–127. MR 1628068
- Bo Guan and Huai-Yu Jian, The Monge-Ampère equation with infinite boundary value, Pacific J. Math. 216 (2004), no. 1, 77–94. MR 2094582, DOI 10.2140/pjm.2004.216.77
- Cristian E. Gutiérrez, The Monge-Ampère equation, Progress in Nonlinear Differential Equations and their Applications, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1829162, DOI 10.1007/978-1-4612-0195-3
- J. B. Keller, On solutions of $\Delta u=f(u)$, Comm. Pure Appl. Math. 10 (1957), 503–510. MR 91407, DOI 10.1002/cpa.3160100402
- Alan V. Lair, A necessary and sufficient condition for existence of large solutions to semilinear elliptic equations, J. Math. Anal. Appl. 240 (1999), no. 1, 205–218. MR 1728197, DOI 10.1006/jmaa.1999.6609
- A. C. Lazer and P. J. McKenna, Asymptotic behavior of solutions of boundary blowup problems, Differential Integral Equations 7 (1994), no. 3-4, 1001–1019. MR 1270115
- A. C. Lazer and P. J. McKenna, On singular boundary value problems for the Monge-Ampère operator, J. Math. Anal. Appl. 197 (1996), no. 2, 341–362. MR 1372183, DOI 10.1006/jmaa.1996.0024
- Jerk Matero, The Bieberbach-Rademacher problem for the Monge-Ampère operator, Manuscripta Math. 91 (1996), no. 3, 379–391. MR 1416719, DOI 10.1007/BF02567962
- Jerk Matero, Quasilinear elliptic equations with boundary blow-up, J. Anal. Math. 69 (1996), 229–247. MR 1428101, DOI 10.1007/BF02787108
- Ahmed Mohammed, Existence and asymptotic behavior of blow-up solutions to weighted quasilinear equations, J. Math. Anal. Appl. 298 (2004), no. 2, 621–637. MR 2086979, DOI 10.1016/j.jmaa.2004.05.030
- Robert Osserman, On the inequality $\Delta u\geq f(u)$, Pacific J. Math. 7 (1957), 1641–1647. MR 98239
- H. Rademacher, Einige besondere probleme partieller Differentialgleichun (1943), 838-845.
- Paolo Salani, Boundary blow-up problems for Hessian equations, Manuscripta Math. 96 (1998), no. 3, 281–294. MR 1638149, DOI 10.1007/s002290050068
- Kaising Tso, On a real Monge-Ampère functional, Invent. Math. 101 (1990), no. 2, 425–448. MR 1062970, DOI 10.1007/BF01231510
Additional Information
- Ahmed Mohammed
- Affiliation: Department of Mathematical Sciences, Ball State University, Muncie, Indiana 47306
- ORCID: setImmediate$0.27459675662351213$1
- Email: amohammed@bsu.edu
- Received by editor(s): July 25, 2005
- Published electronically: June 20, 2006
- Communicated by: David S. Tartakoff
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 141-149
- MSC (2000): Primary 35J65, 35J60, 35J25
- DOI: https://doi.org/10.1090/S0002-9939-06-08623-0
- MathSciNet review: 2280183