The Asymptotic Behaviour Of The Ground State Solution For Biharmonic Elliptic Equation

In this paper, we consider the asymptotic behaviour of the ground state solution for the following biharmonic elliptic equation:whereΩis a unit ball of Rⁿ, n≥5. 2^*=(2n)/(n-4) is the critical Sobolev cxponentsfor the embedding H₀²(Ω)→L^p(Ω),H₀²(Ω) is standard Sobolev space,Δ=sum from i=1 to n ((?)²)/((?)x_i²)denotes the N-dimensional Laplacian.It proved that for p close to 2^*, the ground state solution u_p concentrates near the boundary ofΩand u_p has a unique maximum point x_p and dist(x_p, (?)Ω)→0 as p→2^*, The asymptotic behaviour of u_p is also given, which deduces that the ground state solution is non-radial.The organization of this paper is as follows:In section 1, we introduce the background associated with biharmonic elliptic equation and the main results of this paper.In section 2,we give some preliminaries of this paper.In section 3, we prove the concentration bahaviour of the ground state solution u_p for problem (1) by the concentration compactness principle.In section 4, we show the asymptotic behaviour of ground state solution u_p by the blow-up technique and deduces u_p is non-radial.