Open Access
May/June 2013 Positive solutions of semilinear elliptic equations: a dynamical approach
Matteo Franca
Differential Integral Equations 26(5/6): 505-554 (May/June 2013). DOI: 10.57262/die/1363266077


This paper is devoted to the study of the structure of positive radial solutions for the following semi-linear equation: $$\Delta u + f(u,|x|)=0 .$$ We require $f$ to be nonnegative and to exhibit both subcritical and supercritical behavior with respect to the Sobolev critical exponent. More precisely we assume that $f$ is subcritical for $u$ small and $|x|$ large and supercritical for $u$ large and $|x|$ small, and we give existence and non-existence results for ground states regular and singular, with either fast or slow decay. We find a surprisingly rich structure, which is characterized by two different patterns of bifurcations. We perform a Fowler transformation and we use a dynamical approach, exploiting some ideas borrowed from Bamon, Del Pino, and Flores, combining them with the use of the translation of the Pohozaev function for this dynamical context.


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Matteo Franca. "Positive solutions of semilinear elliptic equations: a dynamical approach." Differential Integral Equations 26 (5/6) 505 - 554, May/June 2013.


Published: May/June 2013
First available in Project Euclid: 14 March 2013

zbMATH: 1299.35127
MathSciNet: MR3086398
Digital Object Identifier: 10.57262/die/1363266077

Primary: 34B16 , 35B09 , 35J61

Rights: Copyright © 2013 Khayyam Publishing, Inc.

Vol.26 • No. 5/6 • May/June 2013
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