Spinorial Yamabe-type equations and the Bär-Hijazi-Lott invariant

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