Concentration on the boundary and sign-changing solutions for a slightly subcritical biharmonic problem

We consider the fourth-order nonlinear elliptic problem: { Δ ( a ( x ) Δ u ) = a ( x ) | u | p − 2 − ε u in Ω , u = 0 on ∂ Ω , Δ u = 0 on ∂ Ω , where Ω is a smooth, bounded domain in R N with N ≥ 5 . Here, p : = 2 N N − 4 is the Sobolev critical exponent for the embedding H 2 ∩ H 0 1 ( Ω ) ↪ L p ( Ω ) , and a ∈ C 2 ( Ω ‾ ) is a strictly positive function on Ω ‾ . We establish sufficient conditions on the function a and the domain Ω for this problem to admit both positive and sign-changing solutions with an explicit asymptotic profile. These solutions concentrate and blow up at a point on the boundary ∂Ω as ε → 0 . The proofs of the main results rely on the Lyapunov-Schmidt finite-dimensional reduction method.