  In this paper, we prove the existence of unbounded sequence of eigenvalues
for the fractional $p-$Laplacian with weight in $\mathbb{R}^N.$ We also show a
nonexistence result when the weighthas positive integral. In addition, we show
some qualitative properties of the first eigenfunction including a sharp decay
estimate. Finally, we extend the decay result to the positive solutions of a
Schr\"odinger type equation.
