### Abstract

*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 |
---|---|

Journal | Journal d'Analyse Mathematique |

Publication status | Accepted/In press - 17 May 2018 |

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### Cite this

*Journal d'Analyse Mathematique*.

**A semilinear elliptic equation with competing powers and a radial potential.** / Musso, Monica; Pimentel, Juliana.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A semilinear elliptic equation with competing powers and a radial potential

AU - Musso, Monica

AU - Pimentel, Juliana

PY - 2018/5/17

Y1 - 2018/5/17

N2 - We verify the existence of radial positive solutions for the semi-linear equation −∆u = up − V(y)uq, u >0, in RN 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 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 limr→∞V(r) < 0.

AB - We verify the existence of radial positive solutions for the semi-linear equation −∆u = up − V(y)uq, u >0, in RN 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 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 limr→∞V(r) < 0.

M3 - Article

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

SN - 0021-7670

ER -