Concentration of solutions of non-linear elliptic equations involving critical Sobolev exponent

Abstract

In IRⁿ (n ≥ 3), an interesting property of the semi-linear equation

Is that, when K₀ is a positive constant, solutions can concentrate at any point. When K₀ is not a constant, we show that concentration of solutions requires strong conditions on K₀. Through the stereographic projection, the discussion can be extended to Sⁿ, and is related to bubbling, or the blow-up phenomenon.