Existence of entire radial large solutions for a class of Monge-Ampère type equations and systems. (English) Zbl 1454.35145

Summary: This paper is mainly concerned with existence of entire positive radial large solutions for a class of Monge-Ampère type equations:
\[ \det D^2 u (x)-\alpha \Delta u = a (|x|)f(u), \quad x \in \mathbb{R}^N ,\] and systems:
\[ \begin{aligned} \det D^2 u (x)-\alpha \Delta u & = a (|x|) f(v), \quad & x \in \mathbb{R}^N, \\ \det D^2 v (x)-\beta \Delta v & = b (|x|) g(u), \quad & x \in \mathbb{R}^N, \end{aligned}\] where \(\det D^2 u\) is the so-called Monge-Ampère operator, \(\Delta\) is the classical Laplace operator, \(N \geq 2\), \(\alpha, \beta\) are positive constants, \(f, g : [0, \infty) \to [0, \infty)\) are continuous and nondecreasing, and \(a, b : \mathbb{R}^N \to [0, \infty)\) are continuous.


35J60 Nonlinear elliptic equations
35J96 Monge-Ampère equations
35B08 Entire solutions to PDEs
Full Text: DOI Euclid


[1] D.-P. Covei, “A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights”, Acta Math. Sci. Ser. B 37:1 (2017), 47-57. Mathematical Reviews (MathSciNet): MR3581525
Zentralblatt MATH: 1389.35027
Digital Object Identifier: doi:10.1016/S0252-9602(16)30114-X
· Zbl 1389.35027
[2] P. Delanoë, “Radially symmetric boundary value problems for real and complex elliptic Monge-Ampère equations”, J. Differential Equations 58:3 (1985), 318-344. Mathematical Reviews (MathSciNet): MR797314
Zentralblatt MATH: 0564.35044
Digital Object Identifier: doi:10.1016/0022-0396(85)90003-8
· Zbl 0564.35044
[3] A. R. F. de Holanda, “Blow-up solutions for a general class of the second-order differential equations on the half line”, C. R. Math. Acad. Sci. Paris 355:4 (2017), 426-431. Mathematical Reviews (MathSciNet): MR3634680
Digital Object Identifier: doi:10.1016/j.crma.2017.03.003
· Zbl 1370.34060
[4] X. Ji and J. Bao, “Necessary and sufficient conditions on solvability for Hessian inequalities”, Proc. Amer. Math. Soc. 138:1 (2010), 175-188. Mathematical Reviews (MathSciNet): MR2550182
Zentralblatt MATH: 1180.35234
Digital Object Identifier: doi:10.1090/S0002-9939-09-10032-1
· Zbl 1180.35234
[5] Q. Jin, Y. Li, and H. Xu, “Nonexistence of positive solutions for some fully nonlinear elliptic equations”, Methods Appl. Anal. 12:4 (2005), 441-449. Mathematical Reviews (MathSciNet): MR2258318
Zentralblatt MATH: 1143.35322
Digital Object Identifier: doi:10.4310/MAA.2005.v12.n4.a5
Project Euclid: euclid.maa/1175797468
· Zbl 1143.35322
[6] T. Kusano and C. A. Swanson, “Existence theorems for elliptic Monge-Ampère equations in the plane”, Differential Integral Equations 3:3 (1990), 487-493. Mathematical Reviews (MathSciNet): MR1047748
Zentralblatt MATH: 0723.35030
Project Euclid: euclid.die/1371571146
· Zbl 0723.35030
[7] T. Kusano and C. A. Swanson, “Existence theorems for Monge-Ampère equations in \(\mathbb{R}^N\)”, Hiroshima Math. J. 20:3 (1990), 643-650. Mathematical Reviews (MathSciNet): MR1083432
Zentralblatt MATH: 0723.35030
Digital Object Identifier: doi:10.32917/hmj/1206129054
Project Euclid: euclid.hmj/1206129054
· Zbl 0723.35030
[8] T. Kusano and C. A. Swanson, “Entire solutions of real and complex Monge-Ampère equations”, SIAM J. Math. Anal. 22:1 (1991), 157-168. Mathematical Reviews (MathSciNet): MR1080152
Zentralblatt MATH: 0838.35038
Digital Object Identifier: doi:10.1137/0522010
· Zbl 0838.35038
[9] A. V. Pogorelov, Monge-Ampère equations of elliptic type, P. Noordhoff Ltd., Groningen, The Netherlands, 1964. Mathematical Reviews (MathSciNet): MR0180763
Zentralblatt MATH: 0133.04902
· Zbl 0133.04902
[10] Z. · Zbl 1329.35126
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