IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v8y2020i8p1332-d396938.html
   My bibliography  Save this article

Resonant Anisotropic ( p , q )-Equations

Author

Listed:
  • Leszek Gasiński

    (Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30084 Cracow, Poland)

  • Nikolaos S. Papageorgiou

    (Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece)

Abstract

We consider an anisotropic Dirichlet problem which is driven by the ( p ( z ) , q ( z ) ) -Laplacian (that is, the sum of a p ( z ) -Laplacian and a q ( z ) -Laplacian), The reaction (source) term, is a Carathéodory function which asymptotically as x ± ∞ can be resonant with respect to the principal eigenvalue of ( − Δ p ( z ) , W 0 1 , p ( z ) ( Ω ) ) . First using truncation techniques and the direct method of the calculus of variations, we produce two smooth solutions of constant sign. In fact we show that there exist a smallest positive solution and a biggest negative solution. Then by combining variational tools, with suitable truncation techniques and the theory of critical groups, we show the existence of a nodal (sign changing) solution, located between the two extremal ones.

Suggested Citation

  • Leszek Gasiński & Nikolaos S. Papageorgiou, 2020. "Resonant Anisotropic ( p , q )-Equations," Mathematics, MDPI, vol. 8(8), pages 1-21, August.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1332-:d:396938
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/8/8/1332/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/8/8/1332/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    3. Leszek Gasiński & Nikolaos Papageorgiou, 2013. "A pair of positive solutions for the Dirichlet p(z)-Laplacian with concave and convex nonlinearities," Journal of Global Optimization, Springer, vol. 56(4), pages 1347-1360, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dolf Talman & Zaifu Yang, 2012. "On a Parameterized System of Nonlinear Equations with Economic Applications," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 644-671, August.
    2. Subramanian, S.V. & Subramanyam, Malavika A. & Selvaraj, Sakthivel & Kawachi, Ichiro, 2009. "Are self-reports of health and morbidities in developing countries misleading? Evidence from India," Social Science & Medicine, Elsevier, vol. 68(2), pages 260-265, January.
    3. World Bank, 2002. "Costa Rica : Social Spending and the Poor, Volume 1. Summary of Issues and Recommendations with Executive Summary," World Bank Publications - Reports 15330, The World Bank Group.
    4. Emin Karagözoğlu, 2014. "A noncooperative approach to bankruptcy problems with an endogenous estate," Annals of Operations Research, Springer, vol. 217(1), pages 299-318, June.
    5. Hernández-Hernández, M.E. & Kolokoltsov, V.N. & Toniazzi, L., 2017. "Generalised fractional evolution equations of Caputo type," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 184-196.
    6. Simon Levin & Anastasios Xepapadeas, 2021. "On the Coevolution of Economic and Ecological Systems," Annual Review of Resource Economics, Annual Reviews, vol. 13(1), pages 355-377, October.
    7. Juan Moreno-Ternero & Antonio Villar, 2006. "The TAL-Family of Rules for Bankruptcy Problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 27(2), pages 231-249, October.
    8. Lee, Hiro & van der Mensbrugghe, Dominique, 2005. "The impact of the US safeguard measures on Northeast Asian producers: General equilibrium assessments," MPRA Paper 82288, University Library of Munich, Germany.
    9. Hoang Ngoc Tuan, 2015. "Boundedness of a Type of Iterative Sequences in Two-Dimensional Quadratic Programming," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 234-245, January.
    10. Wang, Daojuan & Hain, Daniel S. & Larimo, Jorma & Dao, Li T., 2020. "Cultural differences and synergy realization in cross-border acquisitions," International Business Review, Elsevier, vol. 29(3).
    11. Wulf Gaertner & Richard Bradley & Yongsheng Xu & Lars Schwettmann, 2019. "Against the proportionality principle: Experimental findings on bargaining over losses," PLOS ONE, Public Library of Science, vol. 14(7), pages 1-18, July.
    12. Zhou, H. & Uhlaner, L.M., 2009. "Knowledge Management in the SME and its Relationship to Strategy, Family Orientation and Organization Learning," ERIM Report Series Research in Management ERS-2009-026-ORG, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    13. Turpie, J.K. & Marais, C. & Blignaut, J.N., 2008. "The working for water programme: Evolution of a payments for ecosystem services mechanism that addresses both poverty and ecosystem service delivery in South Africa," Ecological Economics, Elsevier, vol. 65(4), pages 788-798, May.
    14. Tilman Br�ck & Patricia Justino & Philip Verwimp & Andrew Tedesco & Alexandra Avdeenko, 2013. "Measuring Conflict Exposure in Micro-Level Surveys," HiCN Working Papers 153, Households in Conflict Network.
    15. Erik Ansink & Hans-Peter Weikard, 2012. "Sequential sharing rules for river sharing problems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(2), pages 187-210, February.
    16. Koichi Hamada & Asahi Noguchi, 2005. "The Role of Preconceived Ideas in Macroeconomic Policy: Japan's Experiences in the Two Deflationary Periods," Working Papers 908, Economic Growth Center, Yale University.
    17. Jingyi Xue, 2018. "Fair division with uncertain needs," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 51(1), pages 105-136, June.
    18. Dipak R. Pant, 2013. "Managing the global waste in the 21st century: As an anthropologist views it," LIUC Papers in Economics 263, Cattaneo University (LIUC).
    19. Mikhail A. Sokolovskiy & Xavier J. Carton & Boris N. Filyushkin, 2020. "Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach," Mathematics, MDPI, vol. 8(8), pages 1-13, July.
    20. van der Laan, Gerard & Talman, Dolf & Yang, Zaifu, 2011. "Solving discrete systems of nonlinear equations," European Journal of Operational Research, Elsevier, vol. 214(3), pages 493-500, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:8:y:2020:i:8:p:1332-:d:396938. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.