A rearrangement minimization problem related to a nonlinear parametric boundary value problem

This paper deals with a rearrangement minimization problem, which is associated with a nonlinear parametric boundary value problem. When the parameter is positive and less than the principal eigenvalue of the p-Laplacian type operator, we obtain that the nonlinear parametric boundary value problem has a unique solution. We then establish the solvability of the rearrangement minimization problem. Finally, based on Stampacchia truncation method, we establish the regularity property of the solution to the nonlinear boundary value problem, and then we investigate the symmetric property of the solution to the rearrangement minimization problem when the domain is a ball at the origin.