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Mathematical Notes

, Volume 97, Issue 1–2, pp 73–84 | Cite as

Existence of solutions to boundary-value problems for semilinear Δγ differential equations

  • D. T. Luyen
  • N. M. TriEmail author
Article

Abstract

In this paper, we study the existence of weak solutions for the boundary-value problem
$$\Delta _\gamma u + g(x,u) = 0in\Omega u = u_0 on\partial \Omega ,$$
(1)
where Ω is a bounded domain with smooth boundary in ℝ N (N ≥ 2) and Δγ is a subelliptic operator of the type
$$\Delta _\gamma u = \sum\limits_{j = 1}^N {\partial _{x_j } (\gamma _j^2 \partial _{x_j } u)} ,\partial _{x_j } u = \frac{{\partial u}} {{\partial x_j }},\gamma = (\gamma _1 ,\gamma _2 ,...,\gamma _N ) $$
. We use the sub-super solution and variational methods.

Keywords

semilinear degenerate elliptic equation subsolution supersolution variational method boundary-value problem 

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Copyright information

© Pleiades Publishing, Ltd. 2015

Authors and Affiliations

  1. 1.Department of MathematicsHoa Lu UniversityNinh Nhat, Ninh Binh CityVietnam
  2. 2.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

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