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Sharp Existence of Ground States Solutions for a Class of Elliptic Equations with Mixed Local and Nonlocal Operators and General Nonlinearity

Author

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  • Tingjian Luo

    (School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China)

  • Qihuan Xie

    (School of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China)

Abstract

In this paper, we study the existence/non-existence of ground states for the following type of elliptic equations with mixed local and nonlocal operators and general nonlinearity: ( − ▵ ) s u − ▵ u + λ u = f ( u ) , x ∈ R N , which is driven by the superposition of Brownian and Lévy processes. By considering a constrained variational problem, under suitable assumptions on f , we manage to establish a sharp existence of the ground state solutions to the equation considered. These results improve the ones in the existing reference.

Suggested Citation

  • Tingjian Luo & Qihuan Xie, 2023. "Sharp Existence of Ground States Solutions for a Class of Elliptic Equations with Mixed Local and Nonlocal Operators and General Nonlinearity," Mathematics, MDPI, vol. 11(16), pages 1-15, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3464-:d:1214646
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    References listed on IDEAS

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    1. Dipierro, Serena & Valdinoci, Enrico, 2021. "Description of an ecological niche for a mixed local/nonlocal dispersal: An evolution equation and a new Neumann condition arising from the superposition of Brownian and Lévy processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 575(C).
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