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

  • Teresa D'Aprile

    Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy
  • Angela Pistoia

    Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via A. Scarpa 16, 00161 Roma, Italy

Abstract

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

where is a small positive parameter, is a smooth, possibly unbounded, domain, is a superlinear and subcritical nonlinearity, is a positive potential bounded away from zero. No symmetry on or on the domain 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 . 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 .

Cite this article

Teresa D'Aprile, Angela Pistoia, Existence, multiplicity and profile of sign-changing clustered solutions of a semiclassical nonlinear Schrödinger equation. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, pp. 1423–1451

DOI 10.1016/J.ANIHPC.2009.01.002