Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Pliska Studia Mathematica Bulgarica, Vol. 21, No 1, (2012), 47p-70p
We consider the homogeneous Dirichlet problem for a class of equations which generalize the p-Laplace equations as well as the Monge- Amp`ere equations in a strictly convex domain ⊂ Rn, n ≥ 2. In the sub-linear case, we study the comparison between quantities involving the solution to this boundary value problem and the corresponding quantities involving the (radial) solution of a problem in a ball having the same η1- mean radius as . Next, we consider the eigenvalue problem for such a p-Monge-Amp`ere equation and study a comparison between its principal eigenvalue (eigenfunction) and the principal eigenvalue (eigenfunction) of the corresponding problem in a ball having the same η1-mean radius as . Symmetrization techniques and comparison principles are the main tools used to get our results.