Multi-bump standing waves with a critical frequency for nonlinear Schrödinger equations

In this paper we study the existence and qualitative property of standing wave solutions <f>ψ(x,t)=e-iEt/ℎu(x)</f> for the nonlinear Schrödinger equation <f>iℎ∂ψ/∂t+ℎ2/2mΔψ-W(x)ψ+|ψ|p-1ψ=0</f> with <f>E</f> being a critical frequency in the sense that <f>infx∈RN W(x)=E</f>. We show that if the zero set of <f>W-E</f> has several isolated connected components <f>Zi(i=1,…,m)</f> such that the interior of <f>Zi</f> is not empty and <f>∂Zi</f> is smooth, then for <f>ℎ>0</f> small there exists, for any integer <f>k,1⩽k⩽m</f>, a standing wave solution which is trapped in a neighborhood of <f>⋃j=1k Aj</f>, where <f>{Aj | j=1,…,k}</f> is any given subset of <f>{Zi | i=1,…,m}</f>. Moreover the amplitude of the standing wave is of the level <f>ℎ2/p-1</f>. This extends the result of Byeon and Wang (Arch. Rational Mech. Anal. 165 (2002) 295) and is in striking contrast with the non-critical frequency case <f>infx∈RN W(x)>E</f>, which has been studied extensively in the past 20 years. [Copyright &y& Elsevier]