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A semilinear elliptic equation with competing powers and a radial potential

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Abstract

We verify the existence of radial positive solutions for the semilinear equation

$$-\Delta u=u^{p}-V(y) u^{q}, \quad u>0, \text { in } \mathbb{R}^{N}$$

where N ≥ 3, p is close to p* ≔ (N+ 2)/(N − 2), and V is a radial smooth potential. If q is super-critical, namely q > p*, we prove that this problem has a radial solution behaving like a superposition of bubbles blowing-up at the origin with different rates of concentration, provided V(0) < 0. On the other hand, if N/(N − 2) < q < p*, we prove that this problem has a radial solution behaving like a super-position of flat bubbles with different rates of concentration, provided limr→∞V(r) < 0.

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Acknowledgements

The first author is supported by FONDECYT Grant 1160135 and Millennium Nucleus Center for Analysis of PDE, NC130017. The second author was supported by FAPESP (Brazil) Grant #2016/04925-7.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Monica Musso

  2. Departamento de Matemáticas, Universidad Católica de Chile, Macul, 782-0436, Chile

    Monica Musso

  3. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, 209210-580, Santo André, SP, Brazil

    Juliana Pimentel

Authors
  1. Monica Musso
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  2. Juliana Pimentel
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Correspondence to Juliana Pimentel.

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Musso, M., Pimentel, J. A semilinear elliptic equation with competing powers and a radial potential. JAMA 140, 283–298 (2020). https://doi.org/10.1007/s11854-020-0089-4

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  • DOI: https://doi.org/10.1007/s11854-020-0089-4

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