On a closed Riemannian spin manifold \((M^n,g,\sigma )\) with positive scalar curvature we consider the spinorial Yamabe-type equation \(D_g\varphi =\lambda |\varphi |^q\varphi \) , where \(\varphi \) is a spinor field, \(\lambda \) is a positive constant and \(q\in (0,\frac{2}{n-1}]\) . For non trivial solutions of this equation we find a positive lower bound for \(\lambda ^2\) . As an application we obtain a conformal lower bound for the Bär-Hijazi-Lott invariant \(\lambda _{min}^+\) when \(q=\frac{2}{n-1}\) . This estimation allow us to get the classification of solutions for spinorial Yamabe equation \(D_g\varphi =\lambda _{min}^+|\varphi |^{2/(n-1)}\varphi \) on manifolds that support a non trivial real Killing spinor. Also we obtain an explicit lower bound for \(\lambda ^2\) when \(q\in (0,\frac{2}{n-1})\) .
