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Matematicheskie Zametki, 2015, Volume 97, Issue 1, paper published in the English version journal
(Mi mzm10917)
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This article is cited in 23 scientific papers (total in 23 papers)
Papers published in the English version of the journal
Existence of Solutions to Boundary-Value Problems for Semilinear $\Delta_{\gamma}$ Differential Equations
D. T. Luyena, N. M. Trib a Department of Mathematics, Hoa Lu University, Ninh Nhat, Ninh Binh City, Vietnam
b Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
Abstract:
In this paper, we study the existence of weak solutions for the boundary-value problem
\begin{equation}
\label{TriLuyen1: DG 1}
\Delta_{\gamma}u+g(x,u)=0 \quad\text{in}\ \ \Omega,\qquad u=u_0 \quad\text{on}\ \ \partial \Omega,
\end{equation}
where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$ ($N \ge 2$) and $\Delta_{\gamma}$ is a subelliptic operator of the type
$$
{{\Delta }_{\gamma }}u=\sum\limits_{j=1}^{N}{{{\partial }_{{{x}_{j}}}}
(\gamma _{j}^{2}{{\partial }_{{{x}_{j}}}}u ),\qquad {{\partial }_{{{x}_{j}}}}u
=\frac{\partial u}{\partial {{x}_{j}}}},\qquad \gamma = (\gamma_1, \gamma_2, \dots, \gamma_N).
$$
We use the sub-super solution and variational methods.
Keywords:
semilinear degenerate elliptic equation, subsolution, supersolution, variational method, boundary-value problem.
Received: 14.05.2014
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