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# A non periodic and asymptotically linear indefinite variational problem in $$\mathbb{R}^N$$

Ponente: Liliane Maia
Tipo de Evento: Investigación
Cuándo 07/04/2016 de 10:00 a 11:00 Salón 2 vCal iCal

A nonlinear Schrödinger equation which models a light beam
propagating in a saturable medium may present a sign changing
potential in the linear term and lead to a semilinear elliptic
equation in $$\mathbb{R}^N$$ with a potential that has a negative
part, see [2].  We will present some recent results on the existence
of nontrivial solution for
\begin{equation*}
\begin{array}{lc} -\Delta u + V(x) u = f(u) & \quad{in} \
\mathbb{R}^N,
\end{array} \tag{$$P$$}
\end{equation*} $$N\geq 3$$, with a non-periodic continuous potential
$$V$$ which may change sign, with an asymptotic limit $$V_\infty$$ at
infinity and a function $$f$$ asymptotically linear at infinity.

We do not use projections on the Nehari manifold either apply the
generalized Nehari method as in [1]. We apply the classical linking
theorem with Cerami condition. This is possible by using the positive
ground state solution $$u_0$$ of limit problem
\begin{equation*} -\Delta u + V_\infty u = f(u) \ \ \quad{in} \ \
\mathbb{R}^N, \tag{$$P_\infty$$}
\end{equation*} projected on a infinite dimensional subspace of
$${H^1(\mathbb{R}^N)}$$ with finite codimension. Moreover, it is
crucial to estimate the interactions of the translates of $$u_0$$ in
order to obtain the linking geometry. Furthermore, the lack of
compactness due to working with a problem in the unbounded domain
$$\mathbb{R}^N$$ is circumvent by an assumption of a spectral gap of
the operator $$-\Delta + V$$.

This is a work in collaboration with José Carlos de Oliveira
Jr. and Ricardo Ruviaro (UnB, Brazil).

[1] A. A. Pankov,  Periodic nonlinear
Schrödinger equation with application to photonic crystals,
Milan J. Math., 73 (2005), 259--287.

[2] C. A. Stuart,  Guidance properties of nonlinear
planar waveguides,  Arch. Rational Mech. Anal.,
125 (1993), 145--200.