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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
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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 Society – Springer 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

  • Existence theorems for some nonlinear fourth-order elliptic boundary value problems
    Alexiades, V; Elcrat, RA; Schaefer, WP
  • Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation
    An, Y; Liu, R
  • On a fourth-order elliptic equation with critical nonlinearity in dimension six
    Ayed, BM; Hammami, M
  • Multiplicity of solutions for a fourth-order elliptic equation with critical exponent on compact manifolds
    Benalili, M
  • An existence criterion for the critical points on unbounded manifolds
    Cerami, G
  • On the existence of eigenvalues for a nonlinear boundary value problem
    Cerami, G
  • An embedding theorem for Sobolev spaces related to nonsmooth vector fields and Harnack inequality
    Franchi, B; Lanconelli, E
  • Existence and nonexistence results for semilinear equations on the Heisenberg group
    Garofalo, N; Lanconelli, E
  • A certain class of hypoelliptic operators
    Grushin, VV
  • Hypoelliptic second order differential equations
    Hörmander, L
  • The Poincaré inequality for vector fields satisfying Hörmander’s condition
    Jerison, D
  • The Yamabe problem on CR manifolds
    Jerison, D; Lee, MJ
  • On the analyticity of solutions to semilinear differential equations degenerated on a submanifold
    Khanh, TT; Tri, NM
  • On semilinear Δλ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _\lambda -$$\end{document}Laplace equation
    Kogoj, AE; Lanconelli, E
  • Existence of nontrivial solution for fourth-order semilinear 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 in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^N$$\end{document}
    Luyen, DT
  • Existence of infinitely many solutions for semilinear degenerate Schrödinger equations
    Luyen, DT; Tri, NM
  • Multiple solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations
    Luyen, DT; Cuong, PV
  • On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators
    Luyen, DT; Tri, NM
  • Infinitely many solutions for a class of perturbed degenerate elliptic equations involving the Grushin operator
    Luyen, DT; Tri, NM
  • Fourth-order Navier boundary value problem with combined nonlinearities
    Pu, Y; Wu, P; Tang, LC
  • On weak solutions of ultraparabolic equations
    Ragusa, MA
  • Hypoelliptic differential operators and nilpotent groups
    Rothschild, PL; Stein, ME
  • Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations
    Thuy, PT; Tri, NM
  • Some existence and nonexistence results for boundary value problems for semilinear elliptic degenerate operators
    Thuy, NTC; Tri, NM
  • On the Grushin equation
    Tri, NM
  • Semilinear perturbation of powers of the Mizohata operators
    Tri, NM
  • Semilinear hypoelliptic differential operators with multiple characteristics
    Tri, NM
  • Multiplicity of solutions for a class of fourth-order elliptical equations
    Wang, W; Zang, W; Zhao, P
  • Multiplicity results for some fourth-order elliptic equations
    Wei, HY
  • Existence of solutions for some fourth-order nonlinear elliptical equations
    Yang, Y; Zhang, HJ
  • Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems
    Zhang, J; Wei, Z
  • Geometry of Grushin spaces
    Wu, MJ