Asymptotic Behavior of Ground State Solution to Hénon Type System

Consider the following Hénon type system: 0.1 {[ -Δ u = |x|^α v^p, in B_1(0),; -Δ v = |x|^α u^q, in B_1(0),; u = v= 0, on ∂ B_1(0), ]. where B_1(0)⊂ℝ^N is the unit ball centered at the origin, N≥ 3 , α > 0 and p, q>1 satisfy: 1p+1 + 1q+1 > N-2N. It is shown that the ground state solution of Eq. 0.1 exists for each α >0 and satisfies the foliated Schwartz symmetry property when α is large. In this paper, we first investigate the asymptotic behavior of the ground state solution ( u_α , v_α ) for Eq. 0.1, as α→ +∞ . Then we analyse the profile of the solutions with one peak or multiple peaks.