Linear and semilinear elliptic equations with a singular potential

This dissertation is concerned with simple elliptic partial differential equations of the form -Du=Fx,u inW u>0inW u= 0on6W, where $W$ is a smooth bounded domain of $Rn$ and F can depend nonsmoothly on the variable x. In this setting, uniqueness, existence and regularity results of the standard theory may fail, even in the linear case. An educating example is the so-called inverse-square potential with a power nonlinearity, i.e., when Fx,u=c x²u+u ^p+l, where c, $l$ > 0 and p > 1. We show that existence of solutions depends highly on the values of the parameters. Optimal regularity, uniqueness and stability results are also considered. For the general case, we first look at linear right-hand sides Fx,u=ax u+bx and obtain important comparison principles, which enable us in the general case to obtain a sharp criterion of existence for a wide class of nonlinear F.