A semilinear elliptic equation with competing powers and a radial potential

Monica Musso; Juliana Pimentel

doi:10.1007/s11854-020-0089-4

A semilinear elliptic equation with competing powers and a radial potential

Monica Musso, Juliana Pimentel

Department of Mathematical Sciences

Federal University of ABC

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Abstract

We verify the existence of radial positive solutions for the semilinear equation (Formula presented.) 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 lim _r→∞V(r) < 0.

Original language	English
Pages (from-to)	283-298
Number of pages	16
Journal	Journal d'Analyse Mathematique
Volume	140
Issue number	1
DOIs	https://doi.org/10.1007/s11854-020-0089-4
Publication status	Published - 1 Mar 2020

Bibliographical note

Funding Information:
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.

Publisher Copyright:
© 2020, The Hebrew University of Jerusalem.

ASJC Scopus subject areas

Analysis
General Mathematics

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10.1007/s11854-020-0089-4

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Cite this

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abstract = "We verify the existence of radial positive solutions for the semilinear equation (Formula presented.) 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 lim r→∞V(r) < 0. ",

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AB - We verify the existence of radial positive solutions for the semilinear equation (Formula presented.) 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 lim r→∞V(r) < 0.

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A semilinear elliptic equation with competing powers and a radial potential

Abstract

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Concentration phenomena in nonlinear analysis

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A semilinear elliptic equation with competing powers and a radial potential

Abstract

Bibliographical note

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Concentration phenomena in nonlinear analysis

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