One-bubble nodal blow-up for asymptotically critical stationary Schr\"odinger-type equations

We investigate in this work families (u_) >₀ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schr\"odinger equations of the following type: g u_ + h_ u_ = |u_|^p_-2 u_ in a closed manifold (M, g), where h_ converges to h in C¹ (M). Assuming that (u_) >₀ blows-up as a single sign-changing bubble, we obtain necessary conditions for blow-up that constrain the localisation of blow-up points and exhibit a strong interaction between h, the geometry of (M, g) and the bubble itself. These conditions are new and are a consequence of the sign-changing nature of u_.