Existence of nontrivial radial solutions for a class of augmented Monge-Ampère equation

Abstract

In this paper, we prove the uniqueness of nontrivial radially convex solutions to the augmented Monge-Ampère equation via sharp estimates, where the augmented term is specified as negative. First, we transform the equation into an equivalent form more amenable to analysis. By constructing sharp a priori estimates on upper and lower bounds, we derive tight constraints on the solution, which serve as the foundation for proving uniqueness. The uniqueness proof relies on reduction to absurdity: assuming two distinct nontrivial radially convex solutions, we apply the established sharp estimates to them. Through detailed comparison, we show that the discrepancy between the solutions violates the estimate bounds, leading to a contradiction and confirming uniqueness. This result advances the theoretical framework of the Monge-Ampère equation and provides a rigorous basis for its applications. For instance, in geometric modeling and optimal transport, solution uniqueness ensures model stability and reliability, facilitating more accurate predictions. Moreover, our approach inspires new methodologies for studying other augmented nonlinear PDEs, offering transferable analytical techniques.