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Fundamental properties and asymptotic shapes of the singular and classical radial solutions for supercritical semilinear elliptic equations

Abstract

We study singular radial solutions of the semilinear elliptic equation \(\Delta u + f(u) = 0\) on finite balls in \(\mathbf{R}^N\) with \(N \ge 3\). We assume that f satisfies either \(f(u) = u^p+o(u^p)\) with \(p > (N+2)/(N-2)\) or \(f(u) = e^u+ o(e^u)\) as \(u \rightarrow \infty \). We provide the existence and uniqueness of the singular radial solution, and show the convergence of regular radial solutions to the singular solution. Some applications to the bifurcation diagram of an elliptic Dirichlet problem are also given. Our results generalize and improve some known results in the literature.

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Acknowledgements

The first author was supported by JSPS KAKENHI Grant Number JP16K05225 and the second author was supported by JSPS KAKENHI Grant Number 17K05333. This work was also supported by Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.

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Correspondence to Yūki Naito.

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Miyamoto, Y., Naito, Y. Fundamental properties and asymptotic shapes of the singular and classical radial solutions for supercritical semilinear elliptic equations. Nonlinear Differ. Equ. Appl. 27, 52 (2020). https://doi.org/10.1007/s00030-020-00658-4

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  • DOI: https://doi.org/10.1007/s00030-020-00658-4

Keywords

  • Semilinear elliptic equation
  • Singular solution
  • Supercritical
  • Uniqueness
  • Existence
  • Asymptotic shape

Mathematics Subject Classification

  • 35J61
  • 35A05
  • 35A24