Qualitative Properties and Symmetry of Quasilinear Elliptic Equations

PhD Synopsis

Abstract

In this talk we will primarily focus on quasilinear elliptic equations which are Euler-Lagrange equations of Sobolev Inequalities in Euclidean or Hyperbolic space. In one of my work, we have established sharp decay estimates for solutions to the quasilinear Hardy-Sobolev-Maz'ya equation in the Euclidean space. As an application of such estimates we establish the existence of positive solutions to a Brezis-Nirenberg problem with sub-linear perturbations of Hardy-Sobolev equation in a smooth bounded domain containing the origin under a suitable notion of positivity for the perturbations. In another work, we have established sharp decay estimates of positive solutions to the Euler-Lagrange equation of the Poincare-Sobolev inequality in Hyperbolic space and established Hyperbolic symmetry of the same by employing the Alexandrov moving plane method. We also discuss some existence and non-existence results for the same.