### Abstract

We verify the existence of radial positive solutions for the semi-linear equation −∆

*u*=*u*−^{p}*V(y)u*>0, in R^{q}, u*where*^{N }*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 super-position 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 lim_{r→∞}*V*(*r*) < 0.Original language | English |
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Pages (from-to) | 283-298 |

Journal | Journal d'Analyse Mathematique |

Volume | 140 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Mar 2020 |

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### Cite this

Musso, M., & Pimentel, J. (2020). A semilinear elliptic equation with competing powers and a radial potential.

*Journal d'Analyse Mathematique*,*140*(1), 283-298. https://doi.org/10.1007/s11854-020-0089-4