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A positive solution for an asymptotically linear elliptic problem on ℝ 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 ℝ 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 ℝ 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 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 to regain...

Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Silvia Cingolani;  Louis Jeanjean;  Simone Secchi — 2009

ESAIM: Control, Optimisation and Calculus of Variations

In this work we consider the magnetic NLS equation ( ℏ i ∇ - A ( x ) ) 2 u + V ( x ) u - f ( | u | 2 ) u = 0 in ℝ N ( 0 . 1 ) where N ≥ 3 , A : ℝ N → ℝ N is a magnetic potential, possibly unbounded, V : ℝ N → ℝ is a multi-well electric potential, which can vanish somewhere, f is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : ℝ N → ℂ to (0.1), under conditions on the nonlinearity which are nearly optimal.

Multi-peak solutions for magnetic NLS equations without non-degeneracy conditions

Silvia Cingolani;  Louis Jeanjean;  Simone Secchi — 2008

ESAIM: Control, Optimisation and Calculus of Variations

In this work we consider the magnetic NLS equation ( ℏ i ∇ - A ( x ) ) 2 u + V ( x ) u - f ( | u | 2 ) u = 0 in ℝ N ( 0 . 1 ) where N ≥ 3 , A : ℝ N → ℝ N is a magnetic potential, possibly unbounded, V : ℝ N → ℝ is a multi-well electric potential, which can vanish somewhere, is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution u : ℝ N → ℂ to (0.1), under conditions on the nonlinearity which are nearly optimal.

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