  We consider on a closed Riemannian spin manifold $(M^n,g,\sigma)$ the
spinorial Yamabe type equation
$D_g\varphi=\lambda|\varphi|^{\frac{2}{n-1}}\varphi$, where $\varphi$ is a
spinor field and $\lambda$ is a positive constant. For a normalized solution
$\varphi$ of this equation we find a positive lower bound for $\lambda^2$. As
an application we obtain an explicit lower bound of the B\"ar-Hijazi-Lott
invariant for some spin manifolds with positive scalar curvature.
