Convex radial solutions for Monge-Ampère equations involving the gradient

Abstract

This paper deals with the existence and multiplicity of convex radial solutions for the Monge-Amp$ \grave{\text e} $re equation involving the gradient $ \nabla u $: $ \begin{cases} \det (D^2u) = f(|x|, -u, |\nabla u|), x\in B, \\ u|_{\partial B} = 0, \end{cases} $ where $ B: = \{x\in \mathbb R^N: |x| < 1\} $. The fixed point index theory is employed in the proofs of the main results.

Keywords

• monge-ampère equation
• convex radial solution
• krein-rutman theorem
• fixed point index
• a priori estimate