Existence results for the [formula omitted]-Hessian type system with the gradients via [formula omitted]-monotone matrices

Abstract

In this work, we deal with the k-Hessian type system with the gradients SkσD2ui+α|∇ui|I=φi(|x|,−u1,−u2,…,−un), inΩ,ui=0, on∂Ω,i=1,2,…,n,where α≥0, n≥2, 1≤k≤N is a positive integer, I is the identity matrix and Ω stands for the open unit ball in RN(N≥2). Based on appropriate assumptions about φi(i=1,2,…,n), some results regarding existence of negative k-convex radial solution are established. More precisely, at least one solution and at least two solutions are obtained via the R+n-monotone matrices and the fixed point theory. Some basic inequality techniques such as Jensen inequality are applied, which allows us to overcome the difficulty that the k-Hessian type operator is related to the gradient terms. Finally, several examples are provided to show the validity of our main results.