Non-degeneracy and existence of new solutions for the Schr\"odinger equations

TL;DR

This paper investigates the nonlinear Schrödinger equation with a potential, proving the non-degeneracy of multi-bump solutions and using this to construct new solutions in the context of positive, radially symmetric potentials.

Contribution

It establishes the non-degeneracy of multi-bump solutions in a symmetric space and leverages this to generate new solutions for the nonlinear Schrödinger equation.

Findings

Multi-bump solutions are non-degenerate in a symmetric space.

New solutions are constructed based on the non-degeneracy result.

The results apply to potentials V(r) > 0 with specific growth conditions.

Abstract

We consider the following nonlinear problem $$ (P) \quad \quad - \Delta u + V(|y|)u=u^{p},\quad u>0 \quad \mbox{in} \ {\mathbb{R}}^N, \quad u \in H^1({\mathbb{R}}^N), $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that the multi-bump solutions constructed in [20] is non-degenerate in a suitable symmetric space. We also use this non-degenerate result to construct new solutions for (P).