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The sharp exponent in the study of the nonlocal Hénon equation in \({\mathbb {R}}^{N}\): a Liouville theorem and an existence result

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Abstract

We consider the nonlocal Hénon equation

$$\begin{aligned} (-\Delta )^s u= |x|^{\alpha } u^{p},\quad {\mathbb {R}}^{N}, \end{aligned}$$

where \((-\Delta )^s\) is the fractional Laplacian operator with \(0<s<1\), \(-2s<\alpha \), \(p>1\) and \(N>2s\). We prove a nonexistence result for positive solutions in the optimal range of the nonlinearity, that is, when

$$\begin{aligned} 1<p<p^*_{\alpha , s}:=\frac{N+2\alpha +2s}{N-2s}. \end{aligned}$$

Moreover, we prove that a bubble solution, that is a fast decay positive radially symmetric solution, exists when \(p=p_{\alpha , s}^{*}\).

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Acknowledgements

B. B. was partially supported by AEI Grant MTM2016-80474-P and Ramón y Cajal fellowship RYC2018-026098-I (Spain). A. Q. was partially supported by Fondecyt Grant No. 1190282 and Programa Basal, CMM. U. de Chile

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38200, La Laguna, Spain

    B. Barrios

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla V-110, Avda. España, 1680, Valparaiso, Chile

    A. Quaas

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  1. B. Barrios
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Correspondence to B. Barrios.

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Communicated by M. Del Pino.

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Barrios, B., Quaas, A. The sharp exponent in the study of the nonlocal Hénon equation in \({\mathbb {R}}^{N}\): a Liouville theorem and an existence result. Calc. Var. 59, 114 (2020). https://doi.org/10.1007/s00526-020-01763-z

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