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Liouville type theorems for two elliptic equations with advections

Volume 122 / 2019

Anh Tuan Duong, Nhu Thang Nguyen, Thi Quynh Nguyen Annales Polonici Mathematici 122 (2019), 11-20 MSC: Primary 35B53, 35J60; Secondary 35B35. DOI: 10.4064/ap180312-20-9 Published online: 11 January 2019

Abstract

We study the elliptic equations $$ -\Delta u+a(x)\cdot\nabla u=f(u)\ \quad\mbox{in }\mathbb R^{N}, $$ where $ N\geq 3 $, the advection term $a(x)$ is a smooth vector field satisfying a certain decay condition and the nonlinearity $f(u)$ is of the form $-u^{-p},\;p \gt 0,$ or $e^u$. We establish Liouville type theorems for the class of positive stable solutions when $f(u)=-u^{-p}$ and for the class of stable solutions when $f(u)=e^u$. In particular, our results improve some results of B. Lai and L. Zhang [Z. Anal. Anwend. 36 (2017), 283–295] and of L. Ma and J. C. Wei [J. Funct. Anal. 254 (2008), 1058–1087].

Authors

  • Anh Tuan DuongDepartment of Mathematics
    Hanoi National University of Education
    136 Xuan Thuy Street
    Cau Giay District, Ha Noi, Vietnam
    e-mail
  • Nhu Thang NguyenDepartment of Mathematics
    Hanoi National University of Education
    136 Xuan Thuy Street
    Cau Giay District, Ha Noi, Vietnam
    e-mail
  • Thi Quynh NguyenFaculty of Fundamental Science
    Hanoi University of Industry
    298 Cau Dien Street
    Bac Tu Liem District, Ha Noi, Vietnam
    e-mail

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