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Normalized solutions on large smooth domains to the Schrödinger equations with potential and combined nonlinearities: The Sobolev critical case

Xiaolu Lin, Yanjun Liu, Zongyan Lv

Abstract

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrodinger equations with mixed nonlinearities. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential. Besides, Our study can be regarded as a Sobolev critical case complement of Bartsch-Qi-Zou (Math Ann 390, 4813-4859, 2024), which has addressed an open problem raised in Bartsch et al.(Commun Partial Differ Equ 46(9):1729--1756,2021).

Normalized solutions on large smooth domains to the Schrödinger equations with potential and combined nonlinearities: The Sobolev critical case

Abstract

In this paper, we consider the existence and multiplicity of prescribed mass solutions to the following nonlinear Schrodinger equations with mixed nonlinearities. The standard approach based on the Pohozaev identity to obtain normalized solutions is invalid as the presence of potential. Besides, Our study can be regarded as a Sobolev critical case complement of Bartsch-Qi-Zou (Math Ann 390, 4813-4859, 2024), which has addressed an open problem raised in Bartsch et al.(Commun Partial Differ Equ 46(9):1729--1756,2021).

Paper Structure

This paper contains 5 sections, 19 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries.
  3. Proof of Theorem \ref{['beta>0-e<0-Omega']}
  4. Proof of Theorem \ref{['beta>0-e>0-Omega']}
  5. Proof of Theorem \ref{['betale0-e>0-Omega']}

Key Result

Theorem 1.1

Suppose that $(V_0)$ and $\beta>0$ hold, and set Then the following hold for $0<\alpha<\alpha_V$:

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 24 more