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

We consider the fourth-order nonlinear elliptic problem: {Delta(a(x)Delta u) = a(x)|u|p-2-epsilon u in ohm, u=0 on partial derivative ohm, Delta u = 0 on partial derivative ohm, where ohm is a smooth, bounded domain in RN with N >= 5. Here, p := 2N N-4 is the Sobolev critical exponent for the embedding H2 boolean AND H01 (ohm)-* Lp(ohm), and a is an element of C2(ohm) is a strictly positive function on ohm. We establish sufficient conditions on the function a and the domain ohm 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 partial derivative ohm as epsilon-* 0. The proofs of the main results rely on the LyapunovSchmidt finite-dimensional reduction method. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.