Differential and Integral Equations

On local compactness in quasilinear elliptic problems

Khalid Adriouch and Abdallah El Hamidi

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One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work [7], Brézis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relatively compact.

Article information

Differential Integral Equations, Volume 20, Number 1 (2007), 77-92.

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J70: Degenerate elliptic equations 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Adriouch, Khalid; El Hamidi, Abdallah. On local compactness in quasilinear elliptic problems. Differential Integral Equations 20 (2007), no. 1, 77--92. https://projecteuclid.org/euclid.die/1356050281

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