Geodesic
Existence and concentration of semiclassical states for nonlinear Schrödinger equations
Electronic Journal of Differential Equations, Tome 2012 (2012).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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},
     publisher = {mathdoc},
     volume = {2012},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a68/}
}
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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/