Nonexistence of bounded energy solutions for a critical exponent problem

In this paper, we discuss positive solutions for certain weighted elliptic equations with critical Sobolev exponent in . The weights depend on a positive parameter γ, which is allowed to increase to infinity. While for small values of γ solutions are completely classified, an attempt to such a classification is much more difficult for large values of the parameter. In the present work we prove the nonexistence of solutions with bounded energy as γ increases to infinity. We also prove a multiplicity result for high energy solutions.