A class of degenerate elliptic equations and a Dido's problem with respect to a measure

In this paper we consider the following class of linear elliptic problems{−div(A(x)∇u)=xNkexp(−|x|22)f(x)inΩ,u=0on∂Ω∖{xN=0}, where k⩾0k⩾0, Ω   is a domain (possibly unbounded) of R+N={x=(x1,…,xN)∈RN:xN>0}, f   belongs to a suitable weighted Lebesgue space and A(x)=(aij(x))ijA(x)=(aij(x))ij is a symmetric matrix with measurable coefficients satisfyingxNkexp(−|x|22)|ζ|2⩽aij(x)ζiζj⩽CxNkexp(−|x|22)|ζ|2. We compare the solution to such a problem with the solution to a symmetric one-dimensional problem belonging to the same class. Our approach use classical symmetrization methods adapted to a relative isoperimetric inequality with respect to a measure related to the structure of the equation.