Positive solution to Schrödinger equation with singular potential and double critical exponents
type: Item
instance of: scholarly article
title: Positive solution to Schrödinger equation with singular potential and double critical exponents
zbMATH Open document ID: 1468.35042
DOI: 10.4171/RLM/910
author: Yanyan Li
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2021-07-02T00:00:00Z
review text: Summary: In this paper, we consider the following Schrödinger equation with singular potential and double critical exponents: \[ -\Delta u+\frac{A}{|x|^{\alpha}}u=|u|^{2^*-2}u+|u|^{p-2}u+\lambda |u|^{2_{\alpha}^*-2}u,\quad x\in \mathbb{R}^N, \] where \(N\geq 3\), \(\alpha\in(0,2)\), \(p\in(\frac{2N-4+2\alpha}{N-2},2^*)\), and \(A,\lambda>0\) are two real constants, and \(2^*=\frac{2N}{N-2}\) is the Sobolev critical exponent, and \(2_{\alpha}^*=2+\frac{4\alpha}{2N-2-\alpha}\) is the critical exponent with respect to the parameter \(\alpha\). First, using the refined Sobolev inequality, we establish the Lions type theorem. Second, we prove that any nonnegative weak solutions of above equation satisfy Pohozaev type identity. Finally, by using perturbation method, Pohozaev type identity and Lions type theorem, we show the existence of positive solution to above equation. We point out that the double critical exponents is an new phenomenon, and we are the first to consider it. Our result partial extends the results in [\textit{M. Badiale} and \textit{S. Rolando}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 17, No. 1, 1--16 (2006; Zbl 1126.35021)] and [\textit{J. Su} et al., Commun. Contemp. Math. 9, No. 4, 571--583 (2007; Zbl 1141.35056)].
Mathematics Subject Classification ID: 35J10
Mathematics Subject Classification ID: 35Q55
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 7367164
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: singular potential
zbMATH Keywords: critical exponent
zbMATH Keywords: existence of a positive solution
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/rlm/910
OpenAlex ID: W3129979615
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description: scientific article; zbMATH DE number 7367164