A semilinear elliptic equation with competing powers and a radial potential

Monica Musso, Juliana Pimentel

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Abstract

We verify the existence of radial positive solutions for the semi-linear equation −∆u = upV(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.
Original languageEnglish
JournalJournal d'Analyse Mathematique
Publication statusAccepted/In press - 17 May 2018

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Radial Solutions
Semilinear Elliptic Equations
Bubble
Superposition
Positive Radial Solutions
Blowing-up
Semilinear Equations
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A semilinear elliptic equation with competing powers and a radial potential. / Musso, Monica; Pimentel, Juliana.

In: Journal d'Analyse Mathematique, 17.05.2018.

Research output: Contribution to journalArticle

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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.

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