  We generalize Kobayashi's connected-sum inequality to the $\lambda$-Yamabe
invariants. As an application, we calculate the $\lambda$-Yamabe invariants of
$\#m_1\mathbb{RP}^n\# m_2(\mathbb{RP}^{n-1}\times S^1)\#lH^n\#kS_+^n$, for any
$\lambda\in [0,1]$, $n\geq 3$, provided $k+l\geq 1$.
  As a corollary, we prove that $\mathbb{RP}^n$ minus finitely many disjoint
$n$-balls have the same $\lambda$-Yamabe invariants as the hemi-sphere, which
forms an interesting contrast with the famous Bray-Neves results on the Yamabe
invariants of $\mathbb{RP}^3$.
