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

  • Volume 101, Number 5, May, 2017
  • Published:
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Abstract

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 RN (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.

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Authors and Affiliations

  1. Department of Mathematics, Hoa Lu University, Ninh Nhat Ninh Binh city, Vietnam

    D. T. Luyen

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  1. D. T. Luyen
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Correspondence to D. T. Luyen.

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Luyen, D.T. Two nontrivial solutions of boundary-value problems for semilinear Δ γ -differential equations. Math Notes 101, 815–823 (2017). https://doi.org/10.1134/S0001434617050078

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