Differential and Integral Equations

Positive solutions of semilinear elliptic equations: a dynamical approach

Matteo Franca

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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.

Article information

Differential Integral Equations, Volume 26, Number 5/6 (2013), 505-554.

First available in Project Euclid: 14 March 2013

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J61: Semilinear elliptic equations 34B16: Singular nonlinear boundary value problems 35B09: Positive solutions


Franca, Matteo. Positive solutions of semilinear elliptic equations: a dynamical approach. Differential Integral Equations 26 (2013), no. 5/6, 505--554. https://projecteuclid.org/euclid.die/1363266077

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