Nonlinear Schrodinger equation with unbounded or vanishing potentials: Solutions concentrating on lower dimensional spheres

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:252

Article

Bonheure, Denis²  Di Cosmo, Jonathan^1,2  Van Schaftingen, Jean¹ 

[1] Catholic Univ Louvain, Dept Math, B-1348 Louvain, Belgium

[2] Univ Libre Bruxelles, Dept Math, B-1050 Brussels, Belgium

关键词: Stationary nonlinear Schrodinger equation;    Semiclassical states;    Semilinear elliptic problem;    Singular potential;    Vanishing potential;    Radial solution;    Concentration on submanifolds;   

DOI  :  10.1016/j.jde.2011.10.004

来源: Elsevier

【 摘 要 】

We study positive bound states for the equation -epsilon(2) Delta u + V(x)u = K(x)f(u), x is an element of R-N. where epsilon > 0 is a real parameter and V and K are radial positive potentials. We are especially interested in solutions which concentrate on a k-dimensional sphere, 1 <= k <= N - 1, as epsilon -> 0. We adopt a purely variational approach which allows us to consider broader classes of potentials than those treated in previous works. For example, V and K might be singular at the origin or vanish superquadratically at infinity. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free