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Existence of Solution for an Asymptotically Linear Schrödinger-Kirchhoff Equation

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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).

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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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Authors and Affiliations

  1. Departamento de Ciências Exatas e Aplicadas, Universidade Federal do Rio Grande do Norte, Caico, 59300-000, RN, Brazil

    Alex M. Batista

  2. Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil

    Marcelo F. Furtado

Authors
  1. Alex M. Batista
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  2. Marcelo F. Furtado
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Correspondence to Marcelo F. Furtado.

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The authors were partially supported by CNPq/Brazil.

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Batista, A.M., Furtado, M.F. Existence of Solution for an Asymptotically Linear Schrödinger-Kirchhoff Equation. Potential Anal 50, 609–619 (2019). https://doi.org/10.1007/s11118-018-9697-3

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