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Solvability of second-order uniformly elliptic inequalities involving demicontinuous \(\psi \)-dissipative operators and applications to generalized population models

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Abstract

Based on a new developed theory for variational inequalities, the purpose of this article is to investigate existence and uniqueness of nonzero positive weak solutions for a class of general second-order uniformly elliptic inequalities with demicontinuous \(\psi \)-dissipative operators in reflexive smooth Banach spaces and a generalized second-order elliptic inequality, which are often regarded as some conditions allowing the species to survive in biology or ecological context with heterogeneous character. Further, we also discuss existence of eigenvalues for the uniformly elliptic inequality and present a generalized population model arising in ecological context to verify the availability and significance of our main results. Finally, we describe two kinds of open questions, which are future work of our research.

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Acknowledgements

We are very grateful to the anonymous referees and editors for their valuable comments, help and advice to improve our paper.

Funding

This work was partially supported by the Sichuan Science and Technology Program (2019YJ0541), the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (2019QZJ03), the AEI of Spain, project MTM2016-75140-P, co-financed by the European Community fund FEDER, and Xunta de Galicia under grant ED431C 2019/02.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong, Sichuan, 643000, People’s Republic of China

    Heng-you Lan

  2. South Sichuan Center for Applied Mathematics, and Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing, Zigong, Sichuan, 643000, People’s Republic of China

    Heng-you Lan

  3. Instituto de Matematicas, University of Santiago de Compostela, Santiago de Compostela, 15782, Spain

    Juan J. Nieto

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  1. Heng-you Lan
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  2. Juan J. Nieto
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Correspondence to Heng-you Lan.

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Lan, Hy., Nieto, J.J. Solvability of second-order uniformly elliptic inequalities involving demicontinuous \(\psi \)-dissipative operators and applications to generalized population models. Eur. Phys. J. Plus 136, 258 (2021). https://doi.org/10.1140/epjp/s13360-021-01230-4

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