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

TL;DR

This paper studies sign-changing solutions that blow up at a single point for asymptotically critical nonlinear Schrödinger equations on manifolds, revealing new conditions linking geometry, potential, and blow-up behavior.

Contribution

It introduces necessary conditions for single-bubble blow-up solutions, highlighting the influence of sign-changing nature on solution localization and interaction with geometry and potential.

Findings

Necessary conditions for blow-up points are established.

Strong interaction between potential, geometry, and blow-up behavior is demonstrated.

Sign-changing nature imposes new constraints not present in positive solutions.

Abstract

We investigate in this work families $(u_\epsilon)_{\epsilon >0}$ of sign-changing blowing-up solutions of asymptotically critical stationary nonlinear Schr\"odinger equations of the following type: $$\Delta_g u_\epsilon + h_\epsilon u_\epsilon = |u_{\epsilon}|^{p_\epsilon-2} u_\epsilon $$ in a closed manifold $(M,g)$, where $h_\epsilon$ converges to $h$ in $C^1(M)$. Assuming that $(u_\epsilon)_{\epsilon >0}$ blows-up as \emph{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 $u_\epsilon$.