On positive solutions for new class of Schr"odinger equations

Abstract

The aim of this paper is to show the existence of positive solutions of the following Schr\"odinger equations with
potential and indefinite nonlinearity: -\Delta u+u-\mu\frac{u}{|x|^2}=h(x)|u|^{p-2}u \quad x\in\mathbb{R}^3,
where $0<\mu<\frac{1}{4}$, $4<p<6$, $h\in C(\mathbb{R}^3)$, $h$ changes sign in $\mathbb{R}^3.$
Our approach combines variational techniques based on critical point theory and some analysis techniques.