Solution manifolds for some semilinear Schrodinger equations

TL;DR

This paper investigates the existence of solution manifolds for a class of semilinear Schrödinger equations with small parameters, contributing to the understanding of their solution structure under specific conditions.

Contribution

It introduces conditions under which solution manifolds exist for certain semilinear Schrödinger equations with small parameters, advancing the analysis of their solution spaces.

Findings

Existence of solution manifolds established under specific conditions.

Conditions on parameters and functions ensure solution existence.

Provides a framework for analyzing solutions to semilinear Schrödinger equations.

Abstract

In this paper, we study the following semilinear Schr\"odinger system $$ -\triangle u+u=(1+K_\alpha(\epsilon x))|u|^{p-2}u\ in \mathbb{R}^N, u\in H^1(\mathbb{R}^N) $$ where $3\leq p<2^*$ and $\epsilon>0$, $\alpha>0$ are small parameters. Under some conditions on $K_\alpha$ and the parameters $\alpha$ and $\epsilon$, we show that this equation exist solution manifolds.