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Potential Analysis

, Volume 50, Issue 4, pp 609–619 | Cite as

Existence of Solution for an Asymptotically Linear Schrödinger-Kirchhoff Equation

  • Alex M. Batista
  • Marcelo F. FurtadoEmail author
Article
  • 62 Downloads

Abstract

We consider the Kirchhoff equation

$$-\Big(1+\lambda \int |\nabla u|^{2}\Big){\Delta} u+V(x)u=f(u)\quad\text{in }\quad \mathbb{R}^{N}, $$
where N ∈ {3, 4}, λ ≥ 0, the potential V is radial and f can be superlinear or aysmptotically linear at infinity. By using variational methods we obtain, for N = 4, the existence of a ground state radial solution when λ is small. The same holds for N = 3 with no restriction on λ. We also prove that, when λ → 0+, the solutions strongly converge to a solution of −Δu + V (x)u = f(u).

Keywords

Kirchhoff equation Nonlocal problems Radial problems Ground state solution 

Mathematics Subject Classification (2010)

Primary 35J20 Secondary 35J25 35J60 

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Notes

Acknowledgments

The authors would like to thank the referee for his/her useful suggestions which improves the presentation of the paper.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Ciências Exatas e AplicadasUniversidade Federal do Rio Grande do NorteCaicoBrazil
  2. 2.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil

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