Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem
Salomón Alarcón, Jorge Faya, Carolina Rey
TL;DR
This work analyzes the almost critical biharmonic problem with a variable coefficient and Navier boundary conditions on a smooth bounded domain in dimensions $N≥5$. The authors develop a Lyapunov–Schmidt finite-dimensional reduction to construct explicit multi-bubble solutions that concentrate on the boundary as $ε→0$, including both positive and sign-changing profiles. They derive precise reduced-energy expansions that couple boundary geometry, the coefficient $a(x)$, and bubble interactions, and they establish the existence of the two main solution families under natural structural assumptions on $a$ and the domain. The results contribute boundary-concentration phenomena for fourth-order problems and rely on refined boundary Green function estimates, bubble projections, and error analysis to control the high-order interactions at the boundary. The approach yields explicit asymptotic profiles and highlights the delicate interplay between domain geometry, coefficient regularity, and higher-order elliptic dynamics in near-critical regimes.
Abstract
We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} Δ(a(x)Δu) = a(x) \left\vert u \right\vert^{p-2-ε} u \ \text{ in } \ Ω, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial Ω, \hspace{0.6cm} Δu = 0 \ \text{ on } \ \partial Ω, \end{array}\end{equation*} where $Ω$ is a smooth, bounded domain in $\mathbb{R}^N$ with $N \geq 5$. Here, $p := \frac{2N}{N-4}$ is the Sobolev critical exponent for the embedding $H^2 \cap H_0^1(Ω) \hookrightarrow L^p(Ω)$, and $a \in C^2(\overlineΩ)$ is a strictly positive function on $\overlineΩ$. 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 $\partial Ω$ as $ε\to 0$. The proofs of the main results rely on the Lyapunov-Schmidt finite-dimensional reduction method.