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A positive solution for an asymptotically linear elliptic problem on N autonomous at infinity

Louis Jeanjean, Kazunaga Tanaka (2002)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on N . The main difficulties to overcome are the lack of a priori bounds for Palais–Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order...

A positive solution for an asymptotically linear elliptic problem on N autonomous at infinity

Louis Jeanjean, Kazunaga Tanaka (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on N . The main difficulties to overcome are the lack of a priori bounds for Palais–Smale sequences and a lack of compactness as the domain is unbounded. For the first one we make use of techniques introduced by Lions in his work on concentration compactness. For the second we show how the fact that the “Problem at infinity” is autonomous, in contrast to just periodic, can be used in order...

A proof of the stratified Morse inequalities for singular complex algebraic curves using the Witten deformation

Ursula Ludwig (2011)

Annales de l’institut Fourier

The Witten deformation is an analytic method proposed by Witten which, given a Morse function f : M R on a smooth compact manifold M , allows to prove the Morse inequalities. The aim of this article is to generalise the Witten deformation to stratified Morse functions (in the sense of stratified Morse theory as developed by Goresky and MacPherson) on a singular complex algebraic curve. In a previous article the author developed the Witten deformation for the model of an algebraic curve with cone-like singularities...

A Weighted Eigenvalue Problems Driven by both p ( · ) -Harmonic and p ( · ) -Biharmonic Operators

Mohamed Laghzal, Abdelouahed El Khalil, Abdelfattah Touzani (2021)

Communications in Mathematics

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both p ( · ) -Harmonic and p ( · ) -biharmonic operators Δ p ( x ) 2 u - Δ p ( x ) u = λ w ( x ) | u | q ( x ) - 2 u in Ω , u W 2 , p ( · ) ( Ω ) W 0 1 , p ( · ) ( Ω ) , is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces L p ( · ) ( Ω ) and W m , p ( · ) ( Ω ) .

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