Bound state solutions for a class of nonlinear Schrödinger equations

  • Denis Bonheure

    Université libre de Bruxelles, Belgium
  • Jean Van Schaftingen

    Université Catholique de Louvain, Belgium


We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, wherewhereV, Karepositivecontinuousfunctionsandare positive continuous functions andp > 1issubcritical,inaframeworkwhichmayexcludetheexistenceofgroundstates.Namely,thepotentialis subcritical, in a framework which may exclude the existence of ground states. Namely, the potentialVisallowedtovanishatinfinityandthecompetingfunctionis allowed to vanish at infinity and the competing functionKdoesnothavetobebounded.Intheemphsemiclassicallimit,i.e.fordoes not have to be bounded. In the \\emph{semi-classical limit}, i.e. for\varepsilon\sim 0,weprovetheexistenceofboundstatesolutionslocalizedaroundlocalminimumpointsoftheauxiliaryfunction, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function \mathcal{A} = V^\theta K^{-\frac{2}{p-1}},where, where \theta=(p+1)/(p-1)-N/2.Aspecialattentionisdevotedtothequalitativepropertiesofthesesolutionsas. A special attention is devoted to the qualitative properties of these solutions as \varepsilon$ goes to zero.

Cite this article

Denis Bonheure, Jean Van Schaftingen, Bound state solutions for a class of nonlinear Schrödinger equations. Rev. Mat. Iberoam. 24 (2008), no. 1, pp. 297–351

DOI 10.4171/RMI/537