## Differential and Integral Equations

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

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

Frédéric Robert and Kunnath Sandeep

#### 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

**Permanent link to this document**

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