The behavior of the “minimal branch” is investigated for quasilinear eigenvalue problems involving the p-Laplace operator, considered in a smooth bounded domain of 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\not=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.
Castorina, D., Esposito, P., Sciunzi, B. (2009). Degenerate elliptic equations with singular nonlinearities. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 34(3), 279-306 [10.1007/s00526-008-0184-3].
Degenerate elliptic equations with singular nonlinearities
ESPOSITO, PIERPAOLO;
2009-01-01
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 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\not=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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
