In this paper we consider the system in $\mathbb{R}^3$
\begin{equation}
\left\{
\begin{array}{l}
-\varepsilon^2 \Delta u + V(x)u + \phi(x)u = u^p,
\\
-\Delta \phi = u^2,
\end{array}
\right.
\end{equation}
for $p\in (1,5)$. We prove the existence of multi-bump solutions whose bumps
concentrate around a local minimum
of the potential $V(x)$. We point out that such solutions do not exist in the
framework of the usual Nonlinear
Schrödinger Equation.
@article{1296828834,
author = {Ruiz
,
David and Vaira
,
Giusi},
title = {Cluster solutions for the Schr\"odinger-Poisson-Slater problem around a local
minimum of the potential},
journal = {Rev. Mat. Iberoamericana},
volume = {27},
number = {1},
year = {2011},
pages = { 253-271},
language = {en},
url = {http://dml.mathdoc.fr/item/1296828834}
}
Ruiz
,
David; Vaira
,
Giusi. Cluster solutions for the Schrödinger-Poisson-Slater problem around a local
minimum of the potential. Rev. Mat. Iberoamericana, Tome 27 (2011) no. 1, pp. 253-271. http://gdmltest.u-ga.fr/item/1296828834/