Abstract
In this work, we show the existence and multiplicity for the nonlocal Lane-Emden system of the form
where \({\varOmega } \subset \mathbb {R}^{N}\) is a C2 bounded domain, \(\mathbb L\) is a nonlocal operator, ν,τ are Radon measures on Ω, p,q are positive exponents, and ρ,σ > 0 are positive parameters. Based on a fine analysis of the interaction between the Green kernel associated with \(\mathbb L\), the source terms uq,vp and the measure data, we prove the existence of a positive minimal solution. Furthermore, by analyzing the geometry of Palais-Smale sequences in finite dimensional spaces given by the Galerkin type approximations and their appropriate uniform estimates, we establish the existence of a second positive solution, under a smallness condition on the positive parameters ρ,σ and superlinear growth conditions on source terms. The contribution of the paper lies on our unifying technique that is applicable to various types of local and nonlocal operators.
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Acknowledgements
The authors would like to thank the anonymous referees for their helpful comments.
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The authors were supported by Czech Science Foundation, Project GA22-17403S.
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Dedicated to Professor Duong Minh Duc on the occasion of his 70th birthday
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Arora, R., Nguyen, PT. Existence and Multiplicity Results for Nonlocal Lane-Emden Systems. Acta Math Vietnam 48, 3–28 (2023). https://doi.org/10.1007/s40306-022-00485-y
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DOI: https://doi.org/10.1007/s40306-022-00485-y
Keywords
- Nonlocal elliptic systems
- Weak-dual solutions
- Measure data
- Green function
- Multiplicity
- Palais-smale sequences