The role of the Green's function in a non-linear elliptic equation involving the critical Sobolev exponent

On the Location of Blow up Points of Least Energy Solutions to the Brezis-Nirenberg Equation

“…In this direction, Han [5] and Rey [7], [8] proved independently the following result, which Brezis and Peletier [3] had conjectured previously.…”

“…0. For this purpose, we combine the method used by Isobe [6] and technical calculations in Rey [7], [8]. Isobe studied the asymptotic behavior of large solutions ðu H Þ H>0 to H-systems constructed by the method of Brezis and Coron [1] as H !…”

A note on a result of M. Grossi

“…As was made explicit by Druet [9], and earlier by Schoen [21] and Rey [20] (among others), Green's function of the operator −∆u + a(x)u with zero Dirichlet boundary conditions on Ω plays an important role in the existence of solutions. For supercritical exponents, besides the early nonexistence results, there are few positive (existence) results (see e.g.…”

The Ljapunov–Schmidt Reduction for Some Critical Problems

“…Han in [27] proved that these solutions blow-up at a critical point of the Robin's function as ǫ goes to zero. Conversely, Rey in [40,41] proved that any C 1 −stable critical point z0 of the Robin's function generates a family of solutions which blows-up at z0 as ǫ goes to zero. MussoPistoia in [35] and Bahri-Li-Rey in [3] studied existence of solutions which blow-up at κ different points of Ω. Grossi-Takahashi [26] proved the nonexistence of positive solutions blowing up at κ ≥ 2 points for these problems in convex domains.…”