Degenerate elliptic equations with singular nonlinearities

Abstract

The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of \({\mathbb{R}^N}\) , and compactness holds below a critical dimension N #. The nonlinearity f(u) lies in a very general class and the results we present are new even for p = 2. Due to the degeneracy of p-Laplace operator, for p ≠ 2 it is crucial to define a suitable notion of semi-stability: the functional space we introduce in the paper seems to be the natural one and yields to a spectral theory for the linearized operator. For the case p = 2, compactness is also established along unstable branches satisfying suitable spectral information. The analysis is based on a blow-up argument and stronger assumptions on the nonlinearity f(u) are required.

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Correspondence to Pierpaolo Esposito.

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Authors are partially supported by MIUR, project “Variational methods and nonlinear differential equations”.

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Castorina, D., Esposito, P. & Sciunzi, B. Degenerate elliptic equations with singular nonlinearities. Calc. Var. 34, 279–306 (2009). https://doi.org/10.1007/s00526-008-0184-3

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Mathematics Subject Classification (2000)

  • 35B35
  • 35B45
  • 35J70
  • 35J60