## Differential and Integral Equations

- Differential Integral Equations
- Volume 18, Number 5 (2005), 495-508.

### A compactness type result for Paneitz-Branson operators with critical nonlinearity

#### Abstract

Given $(M,g),$ a compact Riemannian manifold of dimension $n \ge 8,$ we consider positive solutions $u_{\alpha} $ of ${\Delta}^2_gu - div_g(A_{\alpha} du) + a_{\alpha} u = u^{2^\sharp-1}$, where $ A_{\alpha}$ is a smooth, symmetric (2,0) tensor and $a_{\alpha}$ a smooth function. Assuming that $ A_{\alpha}$ and $a_{\alpha}$ converge in a suitable sense as ${\alpha} \rightarrow \infty$, we obtain conditions under which the weak limit of $u_{\alpha}$ is nontrivial.

#### Article information

**Source**

Differential Integral Equations, Volume 18, Number 5 (2005), 495-508.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356060182

**Mathematical Reviews number (MathSciNet)**

MR2136976

**Zentralblatt MATH identifier**

1212.35092

**Subjects**

Primary: 58J05: Elliptic equations on manifolds, general theory [See also 35-XX]

Secondary: 35B33: Critical exponents 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.)

#### Citation

Sandeep, K. A compactness type result for Paneitz-Branson operators with critical nonlinearity. Differential Integral Equations 18 (2005), no. 5, 495--508. https://projecteuclid.org/euclid.die/1356060182