Differential and Integral Equations

Sharp solvability conditions for a fourth-order equation with perturbation

Frédéric Robert and Kunnath Sandeep

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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


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