Differential and Integral Equations
- Differential Integral Equations
- Volume 26, Number 5/6 (2013), 505-554.
Positive solutions of semilinear elliptic equations: a dynamical approach
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.
Differential Integral Equations, Volume 26, Number 5/6 (2013), 505-554.
First available in Project Euclid: 14 March 2013
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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