Convex radial solutions for Monge-Ampère equations involving the gradient

Area of Science:

• Partial Differential Equations
• Nonlinear Analysis
• Geometric Analysis

Background:

• The Monge-Ampère equation is a fundamental topic in differential geometry and nonlinear analysis.
• Investigating solutions with specific properties, such as convexity and radial symmetry, is crucial for understanding the equation's behavior.
• The inclusion of the gradient term $|\nabla u|$ introduces additional complexity.

Purpose of the Study:

• To establish the existence and multiplicity of convex radial solutions for the specified Monge-Ampère equation.
• To analyze the impact of the nonlinear term $f(|x|, -u, |\nabla u|)$ on the solution set.
• To explore the application of fixed point index theory in this context.

Main Methods:

• The study employs the fixed point index theory as its primary analytical tool.
• Techniques involve transforming the problem into a fixed point problem for an associated operator.
• Analysis of the properties of the equation within the unit ball $B$ with Dirichlet boundary conditions.

Main Results:

• The paper proves theorems concerning the existence of at least one convex radial solution.
• Conditions are derived for the existence of multiple convex radial solutions.
• The results depend on the properties of the function $f$ and the domain $B$.

Conclusions:

• Fixed point index theory provides a powerful framework for studying degenerate elliptic equations like the Monge-Ampère equation.
• The existence and number of convex radial solutions are intricately linked to the nonlinear term $f$.
• This work contributes to the understanding of nonlinear elliptic PDEs with gradient dependence.