One-bubble nodal blow-up for asymptotically critical stationary Schrödinger-type equations

Abstract

We investigate in this work families ${\left(u_{\epsilon }\right)}_{\epsilon >0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schrödinger equations of the following type:${\Delta }_g{u}_{\epsilon }+h_{\epsilon }{u}_{\epsilon }=|u_{\epsilon }{|}^{p_{\epsilon }-2}{u}_{\epsilon }$ in a closed Riemannian manifold $(M,g)$, where $h_{\epsilon }$ converges to h in $C^1\left(M\right)$ and $p_{\epsilon }\to 2_{-}^{\star }:=\frac{2n}{n-2}$ as $\epsilon \to 0$. Assuming that ${\left(u_{\epsilon }\right)}_{\epsilon >0}$ 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 $u_{\epsilon }$.