Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials

Xue Zhang, Marco Squassina, Jianjun Zhang

Abstract

We get multiplicity of normalized solutions for the fractional Schrödinger equation $$ (-Δ)^su+V(\varepsilon x)u=λu+h(\varepsilon x)f(u)\quad \mbox{in $\mathbb{R}^N$}, \qquad\int_{\mathbb{R}^N}|u|^2dx=a, $$ where $(-Δ)^s$ is the fractional Laplacian, $s\in(0,1)$, $a,\varepsilon>0$, $λ\in\mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier, $V,h:\mathbb{R}^N\rightarrow[0,+\infty)$ are bounded and continuous, and $f$ is continuous function with $L^2$-subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of $h$ when $\varepsilon$ is small enough.

Multiplicity of normalized solutions for the fractional Schrödinger equation with potentials

Abstract

We get multiplicity of normalized solutions for the fractional Schrödinger equation where is the fractional Laplacian, , , is an unknown parameter that appears as a Lagrange multiplier, are bounded and continuous, and is continuous function with -subcritical growth. We prove that the numbers of normalized solutions are at least the numbers of global maximum points of when is small enough.
Paper Structure (6 sections, 15 theorems, 152 equations)

This paper contains 6 sections, 15 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. The autonomous problem
  5. The non-autonomous problem
  6. Proof of Theorem \ref{['Th1']}

Key Result

Theorem 1.1

Suppose $(A_1),(A_2)$, $(f_1)-(f_3)$ hold, then there exists $\varepsilon_1>0$ such that problem 1.2 admits at least $k$ couples $(u_j,\lambda_j)\in H^s(\mathbb R^N)\times\mathbb R$ of weak solutions for $\varepsilon\in(0,\varepsilon_1)$ with $\int_{\mathbb R^N}|u_j|^2\mathrm{d}x=a$, $\lambda<0$ and

Theorems & Definitions (30)

  • Theorem 1.1
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 20 more