Non-degeneracy of double-tower solutions for nonlinear Schr\"odinger equation and applications

TL;DR

This paper proves the non-degeneracy of double-tower solutions for a class of nonlinear Schrödinger equations and uses this to establish the existence of new solutions, advancing understanding of solution structures in nonlinear PDEs.

Contribution

It establishes the non-degeneracy of double-tower solutions using Pohozaev identities and blow-up analysis, enabling the discovery of new solution types for the equation.

Findings

Non-degeneracy of double-tower solutions proved

Existence of new solution types demonstrated

Application of Pohozaev identities and blow-up analysis

Abstract

This paper is concerned with the following nonlinear Schr\"odinger equation \begin{equation} \label{eq} - \Delta u + V(|y|)u=u^{p},\quad u>0 \ \ \mbox{in} \ \mathbb {R}^N, \ \ \ u \in H^1(\mathbb {R}^N), \end{equation} where $V(|y|)$ is a positive function, $1<p <\frac{N+2}{N-2}$. Based on the local Pohozaev identities and blow-up analysis, we first prove a non-degeneracy result for double-tower solutions constructed in [18] in a suitable symmetric space. As an application, we obtain the existence of new type solutions for (0.1).