Mathematical Notes

, Volume 101, Issue 5–6, pp 815–823 | Cite as

Two nontrivial solutions of boundary-value problems for semilinear Δ γ -differential equations

  • D. T. LuyenEmail author
Volume 101, Number 5, May, 2017


In this paper, we study the existence of multiple solutions for the boundary-value problem
$${\Delta _\gamma }u + f\left( {x,u} \right) = 0in\Omega ,u = 0on\partial \Omega ,$$
where Ω is a bounded domain with smooth boundary in R N (N ≥ 2) and Δ γ is the subelliptic operator of the type
$${\Delta _\gamma }u = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}u} \right)} ,{\partial _{{x_j}}}u = \frac{{\partial u}}{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \ldots ,{\gamma _N}} \right).$$
We use the three critical point theorem.


Semilinear degenerate elliptic equations critical points two solutions multiple solutions 


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© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Department of MathematicsHoa Lu UniversityNinh Nhat Ninh Binh cityVietnam

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