Schrödinger equations with vanishing potentials involving Brezis-Kamin type problems

We prove the existence of a bounded positive solution for the following stationary Schrodinger equation \begin{document}$ \begin{equation*} -\Delta u+V(x)u = f(x,u),\,\,\, x\in\mathbb{R}^n,\,\, n\geq 3, \end{equation*} $\end{document} where \begin{document}$ V $\end{document} is a vanishing potential and \begin{document}$ f $\end{document} has a sublinear growth at the origin (for example if \begin{document}$ f(x,u) $\end{document} is a concave function near the origen). For this purpose we use a Brezis-Kamin argument included in [ 6 ]. In addition, if \begin{document}$ f $\end{document} has a superlinear growth at infinity, besides the first solution, we obtain a second solution. For this we introduce an auxiliar equation which is variational, however new difficulties appear when handling the compactness. For instance, our approach can be applied for nonlinearities of the type \begin{document}$ \rho(x)f(u) $\end{document} where \begin{document}$ f $\end{document} is a concave-convex function and \begin{document}$ \rho $\end{document} satisfies the \begin{document}$ \mathrm{(H)} $\end{document} property introduced in [ 6 ]. We also note that we do not impose any integrability assumptions on the function \begin{document}$ \rho $\end{document} , which is imposed in most works.