Geodesic
Existence and concentration of semiclassical states for nonlinear Schrödinger equations
Electronic journal of differential equations, Tome 2012 (2012)
In this article, we study the semilinear Schrodinger equation

$ -\epsilon^2\Delta u+ u+ V(x)u=f(u),\quad u\in H^1(\mathbb{R}^N), $

where $N\geq 2$ and $\epsilon>0$ is a small parameter. The function $V$ is bounded in $\mathbb{R}^N, \inf_{\mathbb{R}^N}(1+V(x))>0$ and it has a possibly degenerate isolated critical point. Under some conditions on f, we prove that as $\epsilon\to 0$, this equation has a solution which concentrates at the critical point of V.
Classification : 35J20, 35J70
Keywords: semilinear Schrödinger equation, variational reduction method 
@article{EJDE_2012__2012__a68,
     author = {Chen,  Shaowei},
     title = {Existence and concentration of semiclassical states for nonlinear {Schr\"odinger} equations},
     journal = {Electronic journal of differential equations},
     year = {2012},
     volume = {2012},
     zbl = {1259.35076},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a68/}
}
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TI  - Existence and concentration of semiclassical states for nonlinear Schrödinger equations
JO  - Electronic journal of differential equations
PY  - 2012
VL  - 2012
UR  - http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a68/
LA  - en
ID  - EJDE_2012__2012__a68
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%0 Journal Article
%A Chen,  Shaowei
%T Existence and concentration of semiclassical states for nonlinear Schrödinger equations
%J Electronic journal of differential equations
%D 2012
%V 2012
%U http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a68/
%G en
%F EJDE_2012__2012__a68
Chen,  Shaowei. Existence and concentration of semiclassical states for nonlinear Schrödinger equations. Electronic journal of differential equations, Tome 2012 (2012). http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a68/