Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior

Abstract

In this paper, the equations and systems of Monge-Ampère with parameters are considered. We first show the uniqueness of of nontrivial radial convex solution of Monge-Ampère equations by using sharp estimates. Then we analyze the existence and nonexistence of nontrivial radial convex solutions to Monge-Ampère systems, which includes some new ingredients in the arguments. Furthermore, the asymptotic behavior of nontrivial radial convex solutions for Monge-Ampère systems is also considered. Finally, as an application, we obtain sufficient conditions for the existence of nontrivial radial convex solutions of the power-type system of Monge-Ampère equations.

Positive solutions of singular multiparameter <i>p</i>-Laplacian elliptic systems

“…blow up at infinity) positive solutions by using the maximum principle. On other related problems to (3) can be found in [2,11,14,13,17,16,22,24,26,32,37,38,40,41,51].…”

Solvability Criterion for a System Arising from Monge–Ampère Equations with Two Parameters

“…In addition, some scholars have studied the existence of nontrivial radial convex solutions for a single Monge-Ampère equation or systems of such equations, utilizing the theory of topological degree, bifurcation techniques, the upper and lower solutions method, and so on. For further details, see [2][3][4][5][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and the references therein.…”

Existence and nonexistence of entire k-convex radial solutions to Hessian type system

“…Related to k-Hessian equations, if k = 1 the k-Hessian equations become the well-known Laplacian equations, and if k = N the k-Hessian equations become the Monge-Ampère equations. Concerning Laplacian equations and Monge-Ampère equations, there are a great number of research papers, see for examples [1,6,7,22] and the references therein.…”