Existence and concentration of solutions with compact support for elliptic problems with critical growth in

For a continuous and sign-changing function a ( x ) , we study the existence and asymptotic behavior of compactly supported solutions of the following elliptic problem in R N with N ⩾ 3 : − Δ u = a γ ( x ) u q + u p , u ⩾ 0 and u ∈ D 1 , 2 ( R N ) , where γ > 0 , 0 < q < 1 , p = N + 2 N − 2 and a γ = γ a or a γ = γ a + − a − . When γ is small, both cases admit two solutions v γ and V γ , where v γ is a local minimizer and V γ is a minimizing sequence for the Sobolev embedding from D 1 , 2 ( R N ) to L p + 1 ( R N ) as γ → 0 . In the case a γ = γ a + − a − , V γ blows up in the L ∞ -norm and concentrates at one point as γ → 0 , consequently we show that V γ cannot be radially symmetric when a ( 0 ) < 0 and γ is small enough.