On the weighted rearrangement of functions and degenerate nonlinear elliptic equations

Abstract

Let Ω be a bounded domain of Rⁿ. We shall deal with boundary value problems of the following form −∑_i=1ⁿ ∂ ∕ ∂x_i (|x|^α a_i(x, u, ∇u)) = |x|^α H(x, u) in Ω, u = g on ∂Ω. Here α > 1 - n, u is the relevant solution, ∇u is its gradient and H is a given real-valued function. Under proper assumptions a pri-ori estimates of solutions u to the problem (0.1) are established by virtue of weighted rearrangement of functions and weighted isoperimetric inequalities.