  Consider $N$ bosons in a finite box $\Lambda= [0,L]^3\subset \bR^3$
interacting via a two-body nonnegative soft potential $V= \lambda \tilde V$
with $\tilde V$ fixed and $\lambda>0$ small. We will take the limit $L, N \to
\infty$ by keeping the density $\varrho= N/L^{3}$ fixed and small. We construct
a variational state which gives an upper bound on the ground state energy per
particle $\e$ $$ \e \le 4\pi\varrho a \Big [1+ \frac{128}{15\sqrt{\pi}}(\varrho
a^3)^{1/2}S_\lambda \Big ]
  + O(\varrho^2|\log\varrho|), \quad {as $\varrho\to 0$} $$ with a constant
satisfying $$
  1\leq S_\lambda \leq 1+C\lambda. $$ Here $a$ is the scattering length of $V$
and thus depends on $\lambda$. In comparison, the prediction by Lee-Yang
\cite{LYang} and Lee-Huang-Yang \cite{LHY} asserts that $S_\lambda=1$
independent of $\lambda$.
