Advances in Differential Equations

On a variational problem involving critical Sobolev growth in dimension four

Mokhless Hammami and Mohamed ben Ayed

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In this paper we consider the following nonlinear elliptic problem: $-\Delta u=Ku^3,$ $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $K$ is a positive function and $\Omega$ is a bounded domain of $R^4$. We prove a version of the Morse lemma at infinity for this problem, which allows us to describe the critical points at infinity of the associated variational problem. Using a topological argument, we are able to prove an existence result.

Article information

Adv. Differential Equations, Volume 9, Number 3-4 (2004), 415-446.

First available in Project Euclid: 18 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 35B33: Critical exponents 35J60: Nonlinear elliptic equations 47J30: Variational methods [See also 58Exx] 49J10: Free problems in two or more independent variables 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


ben Ayed, Mohamed; Hammami, Mokhless. On a variational problem involving critical Sobolev growth in dimension four. Adv. Differential Equations 9 (2004), no. 3-4, 415--446.

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