In this paper we establish the existence of a positive solution for an asymptotically linear elliptic problem on . 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...
In this paper we establish the existence of a positive solution
for an asymptotically linear elliptic problem on . 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...
In this work we consider the magnetic NLS equation
where , is a magnetic potential, possibly unbounded, 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 to (0.1), under conditions on the nonlinearity which are nearly optimal.
In this work we consider the magnetic NLS equation
where , is a magnetic potential,
possibly unbounded, 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 to (0.1), under conditions
on the nonlinearity which are nearly optimal.
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