## Differential and Integral Equations

### Sharp solvability conditions for a fourth-order equation with perturbation

#### Abstract

Let $B$ be the unit ball of ${\mathbb{R}^n}$, $n\geq 5$, and $\rho:\mathbb{R}\rightarrow\mathbb{R}$ a smooth function. We consider the following critical problem: $$\left\{\begin{array}{ll} \Delta^2 u=|u|^{\frac{8}{n-4}}u+\rho(u) & \hbox{ in }B\\ u\not\equiv 0 & \\ u=\frac{\partial u}{\partial n}=0 & \hbox{ on }\partial B . \end{array}\right.$$ We give sufficient conditions for the existence of solutions to this problem. These conditions are close to being sharp, as we prove by considering the problem on arbitrary small balls.

#### Article information

Source
Differential Integral Equations, Volume 16, Number 10 (2003), 1181-1214.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356060544

Mathematical Reviews number (MathSciNet)
MR2014806

Zentralblatt MATH identifier
1145.35334

Subjects
Primary: 35J40: Boundary value problems for higher-order elliptic equations
Secondary: 35J60: Nonlinear elliptic equations

#### Citation

Robert, Frédéric; Sandeep, Kunnath. Sharp solvability conditions for a fourth-order equation with perturbation. Differential Integral Equations 16 (2003), no. 10, 1181--1214. https://projecteuclid.org/euclid.die/1356060544