Abstract
This paper considers minimizers of the following inhomogeneous
$L^2$-subcritical energy functional
\[E(u):=\int_{\R^N}|\nabla
u|^{2}dx-\frac{2}{p+1}\int_{\R^N}m(x)|u|^{p+1}dx,%\
u\in H^{1}(\R^N),
\] under the mass constraint
$\|u\|^{2}_{2}=M$.
Here $N\geq1$,
$p\in(1,1+\frac{4}{N})$,
$M>0$ and the inhomogeneous term $m(x)$ satisfies $0