Abstract
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, whereV, Karepositivecontinuousfunctionsandp > 1issubcritical,inaframeworkwhichmayexcludetheexistenceofgroundstates.Namely,thepotentialVisallowedtovanishatinfinityandthecompetingfunctionKdoesnothavetobebounded.Intheemphsemi−classicallimit,i.e.for\varepsilon\sim 0,weprovetheexistenceofboundstatesolutionslocalizedaroundlocalminimumpointsoftheauxiliaryfunction\mathcal{A} = V^\theta K^{-\frac{2}{p-1}},where\theta=(p+1)/(p-1)-N/2.Aspecialattentionisdevotedtothequalitativepropertiesofthesesolutionsas\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