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

Year: 2025
Type: article
Abstract: 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.
DOI: 10.1016/j.jde.2025.113285
Language: English
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W4409309662
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Cites: 34
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Related to: 10