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

TL;DR

This paper investigates boundary-concentrating, sign-changing solutions for a slightly subcritical biharmonic problem, revealing conditions under which solutions blow up at the boundary as a parameter approaches zero.

Contribution

It provides new sufficient conditions on the domain and coefficient function for existence of boundary-concentrating solutions with explicit profiles, using Lyapunov-Schmidt reduction.

Findings

Solutions concentrate and blow up at boundary points as epsilon approaches zero.

Existence of both positive and sign-changing solutions under specified conditions.

Explicit asymptotic profiles of solutions are derived.

Abstract

We consider the fourth-order nonlinear elliptic problem: \begin{equation*} \begin{array}{ll} \Delta(a(x)\Delta u) = a(x) \left\vert u \right\vert^{p-2-\epsilon} u \ \text{ in } \ \Omega, \hspace{0.6cm} u = 0 \ \text{ on } \ \partial \Omega, \hspace{0.6cm} \Delta u = 0 \ \text{ on } \ \partial \Omega, \end{array}\end{equation*} where $\Omega$ 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 $H^2 \cap H_0^1(\Omega) \hookrightarrow L^p(\Omega)$, and $a \in C^2(\overline{\Omega})$ is a strictly positive function on $\overline{\Omega}$. We establish sufficient conditions on the function $a$ and the domain $\Omega$ 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 $\partial \Omega$ as $\epsilon \to 0$. The proofs of the main results rely on the Lyapunov-Schmidt finite-dimensional reduction method.