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

Abstract

We study the existence and multiplicity of sign-changing solutions for the Dirichlet problem

where $\varepsilon$ 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 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 $V$.