## Advances in Differential Equations

### Concentration estimates and multiple solutions to elliptic problems at critical growth

#### Abstract

In this paper, we consider the problem $-\Delta u =|u| ^{2^*-2}u+\lambda u$ in $\Omega$, $u= 0$ on $\partial \Omega$, where $\Omega$ is an open regular bounded subset of $\mathbb R^N$ $(N\geq 3)$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $\lambda>0$. Our main result asserts that, if $N\geq 7$, the problem has infinitely many solutions and, from the point of view of the compactness arguments employed here, the restriction on the dimension $N$ cannot be weakened.

#### Article information

Source
Adv. Differential Equations, Volume 7, Number 10 (2002), 1257-1280.

Dates
First available in Project Euclid: 27 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651637

Mathematical Reviews number (MathSciNet)
MR1919704

Zentralblatt MATH identifier
1208.35048

#### Citation

Devillanova, Giuseppe; Solimini, Sergio. Concentration estimates and multiple solutions to elliptic problems at critical growth. Adv. Differential Equations 7 (2002), no. 10, 1257--1280. https://projecteuclid.org/euclid.ade/1356651637