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Nontrivial solutions to boundary value problems for semilinear Δγ-differential equations

Applications of Mathematics volume 66, pages 461–478 (2021)Cite this article

Abstract

In this article, we study the existence of nontrivial weak solutions for the following boundary value problem:

$$ - {\Delta _\gamma }u = f(x,u)\;{\rm{in}}\;\Omega ,\;\;\;\;\;u = 0\;{\rm{on}}\;\partial \Omega ,$$

where Ω is a bounded domain with smooth boundary in \({\mathbb{R}^N},\;\Omega \cap \left\{ {{x_j} = 0} \right\} \ne \emptyset \) for some j, Δγ is a subelliptic linear operator of the type

$${\Delta _\gamma }: = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}(\gamma _j^2{\partial _{{x_j}}}),\;\;\;\;\;{\partial _{{x_j}}}: = {\partial \over {\partial {x_j}}}} ,\;\;\;\;\;N \ge 2,$$

where γ(x) = (γ1(x), γ2(x), …, γN(x)) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f(x, ξ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.

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Acknowledgements

The author warmly thanks the anonymous referees for the careful reading of the manuscript and for their useful and nice comments.

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Authors and Affiliations

  1. Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, 19 Nguyen Huu Tho street, Tan Phong ward, District 7, Ho Chi Minh City, Vietnam

    Duong Trong Luyen

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, 19 Nguyen Huu Tho street, Tan Phong ward, District 7, Ho Chi Minh City, Vietnam

    Duong Trong Luyen

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  1. Duong Trong Luyen
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Correspondence to Duong Trong Luyen.

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This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2020.13.

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Luyen, D.T. Nontrivial solutions to boundary value problems for semilinear Δγ-differential equations. Appl Math 66, 461–478 (2021). https://doi.org/10.21136/AM.2021.0363-19

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Keywords

  • Δγ-Laplace problem
  • Cerami condition
  • variational method
  • weak solution
  • Mountain Pass Theorem

MSC 2020

  • 35J70
  • 35J20
  • 35J25
  • 35D30