Methods and Applications of Analysis

Volume 28 (2021)

Number 2

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part I

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Multiple solutions for logarithmic Schrödinger equations with critical growth

Pages: 221 – 248

DOI: https://dx.doi.org/10.4310/MAA.2021.v28.n2.a6

Authors

Yinbin Deng (School of Mathematics and Statistics, Central China Normal University, Wuhan, China)

Huirong Pi (School of Mathematics and Information Science, Guangxi University, Nanning, China)

Wei Shuai (School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, China)

Abstract

In this paper, we establish the existence of positive ground state solution and least energy sign-changing solution for the following logarithmic Schrödinger equation\[-\Delta u + V (x) u = u \operatorname{log} u^2 + {\lvert u \rvert}^{2^\ast-2}u, x \in \mathbb{R}^N.\]It is known that the corresponding variational functional is not well defined in $H^1 (\mathbb{R}^N)$. Via direction derivative and constrained minimization method, we first prove the existence of positive ground state solution and least energy sign-changing solution for the following subcritical problem\[-\Delta u + V (x) u = u \operatorname{log} u^2 + {\lvert u \rvert}^{p-2}u, x \in \mathbb{R}^N.\]Then, we analyze the behavior of solutions for subcritical problem and pass the limit as the exponent $p$ approaches to $2^\ast$.

Received 10 November 2020

Accepted 28 January 2021

Published 10 June 2022