Concentration on the Boundary and Sign-Changing Solutions for a Slightly Subcritical Biharmonic Problem

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 $\hat{I}\text{c◯}$ is a smooth, bounded domain in ${\mathbb{R}}^N$ with $N\ge 5$. Here, $p:=\frac{2N}{N-4}$ is the Sobolev critical exponent for the embedding $H^2\cap H_0^1(\hat{I}\text{c◯})\hookrightarrow L^p(\hat{I}\text{c◯})$, and $a\in C^2(\overline{\hat{I}}\text{c◯})$ is a strictly positive function on $\overline{\hat{I}}\text{c◯}$. We establish sufficient conditions on the function $a$ and the domain $\hat{I}\text{c◯}$ 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 $\partial \hat{I}\text{c◯}$ as $\hat{I}\mu \to 0$. The proofs of the main results rely on the Lyapunov-Schmidt finite-dimensional reduction method.