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On stable entire solutions of semilinear elliptic equations with weights
 Proc. Amer. Math. Soc
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Degenerate elliptic equations with singular nonlinearities
 CALCULUS OF VARIATIONS
, 2007
"... The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we prese ..."
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Cited by 7 (7 self)
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The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, 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 pLaplace operator, for p = 2 it is crucial to define a suitable notion of semistability: 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 blowup argument and stronger assumptions on the nonlinearity f (u) are required.
Liouville type theorems for stable solutions of certain elliptic systems
 Adv. Nonlinear Stud
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LOW DIMENSIONAL INSTABILITY FOR SEMILINEAR AND QUASILINEAR PROBLEMS IN RN
"... Abstract. Stability properties for solutions of −∆m(u) = f(u) in RN are investigated, where N ≥ 2 and m ≥ 2. The aim is to identify a critical dimension N # so that every nonconstant solution is linearly unstable whenever 2 ≤ N < N#. For positive, increasing and convex nonlinearities f(u), glob ..."
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Cited by 4 (0 self)
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Abstract. Stability properties for solutions of −∆m(u) = f(u) in RN are investigated, where N ≥ 2 and m ≥ 2. The aim is to identify a critical dimension N # so that every nonconstant solution is linearly unstable whenever 2 ≤ N < N#. For positive, increasing and convex nonlinearities f(u), global bounds on f f (f ′)2 allows us to find a dimension N which is optimal for exponential and power nonlinearities. In the radial setting we can deal more generally with C1−nonlinearities and the dimension N # we find is still optimal.
Calculus of Variations Degenerate elliptic equations with singular nonlinearities
 CALC. VAR.
, 2007
"... The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, and compactness holds below a critical dimension N #. The nonlinearity f (u) lies in a very general class and the results we prese ..."
Abstract
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The behavior of the “minimal branch ” is investigated for quasilinear eigenvalue problems involving the pLaplace operator, considered in a smooth bounded domain of RN, 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 pLaplace operator, for p = 2 it is crucial to define a suitable notion of semistability: 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 blowup argument and stronger assumptions on the nonlinearity f (u) are required.