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 language English Journal d'Analyse Mathematique Accepted/In press - 17 May 2018

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Semilinear Elliptic Equations
Bubble
Superposition
Blowing-up
Semilinear Equations
Verify

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In: Journal d'Analyse Mathematique, 17.05.2018.

Research output: Contribution to journalArticle

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title = "A semilinear elliptic equation with competing powers and a radial potential",
abstract = "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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AU - Pimentel, Juliana

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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.

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JF - Journal d'Analyse Mathematique

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