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Existence and Non-existence of Solutions for Semilinear bi-Δγ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta _{\gamma }-$$\end{document}Laplace Equation

Existence and Non-existence of Solutions for Semilinear bi-Δγ-\documentclass[12pt]{minimal}... In this paper, we study existence and non-existence of weak solutions for semilinear bi-Δγ-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\Delta _{\gamma }-$$\end{document}Laplace equation Δγ2u=f(x,u)inΩ,u=∂νu=0on∂Ω,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \Delta ^2_\gamma u=f(x,u) \ \text { in }\Omega , \quad u= \partial _\nu u =0 \; \text { on }\partial \Omega , \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Omega $$\end{document} is a bounded domain with smooth boundary in RN(N≥2),f(x,ξ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^N \ (N \ge 2), f(x,\xi ) $$\end{document} is a Carathéodory function and Δγ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \Delta _{\gamma }$$\end{document} is the subelliptic operator of the type Δγ:=∑j=1N∂xjγj2∂xj,∂xj:=∂∂xj,γ=(γ1,γ2,...,γN),Δγ2:=Δγ(Δγ).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}}, \gamma = (\gamma _1, \gamma _2, ..., \gamma _N),\quad \Delta ^2_\gamma : =\Delta _\gamma (\Delta _\gamma ). \end{aligned}$$\end{document} http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

Existence and Non-existence of Solutions for Semilinear bi-Δγ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta _{\gamma }-$$\end{document}Laplace Equation

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Publisher
Springer Journals
Copyright
Copyright © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2021
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-021-01223-7
Publisher site
See Article on Publisher Site

Abstract

In this paper, we study existence and non-existence of weak solutions for semilinear bi-Δγ-\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$-\Delta _{\gamma }-$$\end{document}Laplace equation Δγ2u=f(x,u)inΩ,u=∂νu=0on∂Ω,\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \Delta ^2_\gamma u=f(x,u) \ \text { in }\Omega , \quad u= \partial _\nu u =0 \; \text { on }\partial \Omega , \end{aligned}$$\end{document}where Ω\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Omega $$\end{document} is a bounded domain with smooth boundary in RN(N≥2),f(x,ξ)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {R}^N \ (N \ge 2), f(x,\xi ) $$\end{document} is a Carathéodory function and Δγ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ \Delta _{\gamma }$$\end{document} is the subelliptic operator of the type Δγ:=∑j=1N∂xjγj2∂xj,∂xj:=∂∂xj,γ=(γ1,γ2,...,γN),Δγ2:=Δγ(Δγ).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \Delta _\gamma : =\sum \limits _{j=1}^{N}\partial _{x_j} \left( \gamma _j^2 \partial _{x_j} \right) , \quad \partial _{x_j}: =\frac{\partial }{\partial x_{j}}, \gamma = (\gamma _1, \gamma _2, ..., \gamma _N),\quad \Delta ^2_\gamma : =\Delta _\gamma (\Delta _\gamma ). \end{aligned}$$\end{document}

Journal

Bulletin of the Malaysian Mathematical Sciences SocietySpringer Journals

Published: Mar 1, 2022

Keywords: Bi-Δγ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\Delta _{\gamma }-$$\end{document}Laplace equations; Δγ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _\gamma -$$\end{document}Laplace operator; Pohozaev’s type identities; Nontrivial solutions; Weak solutions; Existence; Multiple solutions; Primary 35J35; Secondary 35J50; 35J60

References