On positive entire solutions to a class of equations with a singular coefficient and critical exponent

We prove some results about existence, uniqueness and qualitative behavior of positive solutions to equations of the type $$ -\Delta u=a(x/|x|){u\over |x|^2}+f(x,u)\qquad\;\;\hbox{in }\;\mathbf{R}^n\setminus\{0\}\;,\tag 0.1 $$ depending on the behavior of the function $a$ of the angular variable $x/|x|$. Our main results concern the critical nonlinearity $f(s)=s^{(n+2)/(n-2)}$. The proofs are based on variational arguments and the moving plane method.