LAUSR

Abstract We consider the following slightly subcritical problem ( ℘ e ) { − Δ u = β ( x ) | u | p − 1 − e u in  Ω , u = 0 on  ∂ Ω , where Ω is a smooth bounded domain in R n , 3 ≤ n ≤ 6 , p : = n + 2 n − 2 is the Sobolev critical exponent, e is a small positive parameter and β ∈ C 2 ( Ω ‾ ) is a positive function. We assume that there exists a nondegenerate critical point ξ ⁎ ∈ ∂ Ω of the restriction of β to the boundary ∂Ω such that ∇ ( β ( ξ ⁎ ) − 2 p − 1 ) ⋅ η ( ξ ⁎ ) > 0 , where η denotes the inner normal unit vector on ∂Ω. Given any integer k ≥ 1 , we show that for e > 0 small enough problem ( ℘ e ) has a positive solution, which is a sum of k bubbles which accumulate at ξ ⁎ as e tends to zero. We also prove the existence of a sign changing solution whose shape resembles a sum of a positive bubble and a negative bubble near the point ξ ⁎ .