A limit equation and bifurcation diagrams of semilinear elliptic equations with general supercritical growth

Abstract

We study radial solutions of the semilinear elliptic equation $\Delta u+f(u)=0$ under rather general growth conditions on $f$. We construct a radial singular solution and study the intersection number between the singular solution and a regular solution. An application to bifurcation problems of elliptic Dirichlet problems is given. To this end, we derive a certain limit equation from the original equation at infinity, using a generalized similarity transformation. Through a generalized Cole-Hopf transformation, all the limit equations can be reduced into two typical cases, i.e., $\Delta u+u^p=0$ and $\Delta u+e^u=0$.

21 pages

References in corpus (1)

• Existence and nonexistence of solutions for the heat equation with a superlinear source term

Cited by in corpus (3)

• The Emden-Fowler equation on a spherical cap of ${\mathbb{S}}^N$
• A supercritical scalar field equation with a forcing term
• Fractional semilinear heat equations with singular and nondecaying initial data