Positive multi-bump solutions for the Schrödinger equation with slow decaying competing potentials

We are concerned with the existence of multi-bump solutions to the following nonlinear Schrödinger equation with competing potentials V and Q,$-\Delta u+V\left(\right|x\left|\right)u=Q\left(\right|x\left|\right)u^p,u>0\text{in}{R}^N,$ where $N\ge 3,1<p<\frac{N+2}{N-2}$, V and Q are radial functions having the following slow algebraic decay with $m,n>0$,$V\left(\right|x\left|\right)=V_0+\frac{a}{|x{|}^m}+O\left(\frac{1}{|x{|}^{m+\kappa }}\right),Q\left(\right|x\left|\right)=Q_0+\frac{b}{|x{|}^n}+O\left(\frac{1}{|x{|}^{n+\theta }}\right)\text{ as }\left|x\right|\to \infty \text{,}$ $V_0,Q_0,\kappa ,\theta ,a>0$. By introducing a weighted norm and some delicate analysis, we construct infinitely many new positive multi-bump solutions for $m<n,b\in R$ or $m\ge n,b\le 0$. The maximum points of these bump solutions lie on the top and bottom circles of a cylinder near the infinity. This result complements and extends the existence results of multi-bump solutions in [2], [11] from $m,n>1$ to the slow decaying potentials case $m,n>0$.