Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizers  uA(x)=UA(|x|)uA(x)=UA(|x|), having UA(⋅)UA(⋅)AC monotone   and ℓ∗∗(UA(⋅),0) increasing, for the convex scalar multiple integralequation(∗ )∫BRℓ∗∗(u(x),|∇u(x)|ρ1(|x|))⋅ρ2(|x|)dx among those u(⋅)u(⋅) in the Sobolev space  A+W01,1(BR). Here, |∇u(x)||∇u(x)| is the Euclidean norm of the gradient vector   and BRBR is the ballball {x∈Rd:|x|