Bound state solutions for a class of nonlinear Schrödinger equations

Abstract

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, $where$V, K$arepositivecontinuousfunctionsand$p > 1$issubcritical,inaframeworkwhichmayexcludetheexistenceofgroundstates.Namely,thepotential$V$isallowedtovanishatinfinityandthecompetingfunction$K$$\begin{gathered} doesnothavetobebounded.Inthe \\ emphsemi-classicallimit,i.e.for \end{gathered}$$\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.