Differential and Integral Equations

Sharp solvability conditions for a fourth-order equation with perturbation

Frédéric Robert and Kunnath Sandeep

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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

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

First available in Project Euclid: 21 December 2012

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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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