Existence and asymptotic behavior of strictly convex solutions for singular k-Hessian equations with nonlinear gradient terms

Abstract

Abstract In this paper, we mainly consider the singular k-Hessian equations S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) + g ⁢ ( | D ⁢ u | ) in ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)+g(|Du|)\quad\text{in }\Omega and S k ⁢ ( λ ⁢ ( D 2 ⁢ u ) ) = h ⁢ ( x ) ⁢ f ⁢ ( - u ) ⁢ ( 1 + g ⁢ ( | D ⁢ u | ) ) in ⁢ Ω S_{k}(\lambda(D^{2}u))=h(x)f(-u)(1+g(|Du|))\quad\text{in }\Omega with the Dirichlet boundary condition u = 0 {u=0} on ∂ ⁡ Ω {\partial\Omega} , where Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 2 {N\geq 2} ) is a strictly convex, bounded smooth domain. Using the method of upper and lower solutions and the Karamata regular variation theory, we get new criteria of the existence and asymptotic behavior of strictly convex solutions under different conditions imposed on h, f and g. This problem is more difficult to solve than the k-Hessian problem without gradient terms, and requires additional new conditions in the proof process.