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

We study the existence and multiplicity of sign-changing solutions for the Dirichlet problem{−ε2Δv+V(x)v=f(v)in Ω,v=0on ∂Ω, where ε is a small positive parameter, Ω 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 Ω is assumed. It is known by Kang and Wei (see [X. Kang, J. Wei, On interacting bumps of semiclassical states of nonlinear Schrödinger 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.