Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrodinger equation

We study the existence and multiplicity of sign-changing solutions for the Dirichlet problem {-epsilon(2) Delta v + V(x)v = f(v) in Omega, v = 0 on partial derivative Omega, where epsilon is a small positive parameter, Omega is a smooth, possibly unbounded, domain, f is a superlinear and subcritical nonlinearity, V is a positive potential bounded away from zero. No symmetry on V or on the domain Omega is assumed. It is known by Kang and Wei (see [X. Kang, J. Wei, On interacting bumps of semiclassical states of nonlinear Schrodinger equations, Adv. Differential Equations 5 (2000) 899-928]) that this problem has positive clustered solutions with peaks approaching a local maximum of V. The aim of this paper is to show the existence of clustered solutions with mixed positive and negative peaks concentrating at a local minimum point, possibly degenerate, of V. (C) 2009 Elsevier Masson SAS. All rights reserved.