Positive Solutions for Semilinear Elliptic Equations: Two Simple Models with Several Bifurcations

Abstract

In this paper we analyze the structure of positive radial solutions for the following semi-linear equations:

$$ \Delta u + f(u,|{\bf x}|)=0 $$

where ${{\bf x}\in \mathbb{R}^n}$ and f is superlinear. In fact we just consider two very special non-linearities, i.e.

$$\label{uno} f(u,|{\bf x}|) = u|u|^{q-2}\max\{|{\bf x}|^{\delta^s}, |{\bf x}|^{\delta^u}\}\; \quad -2 < \delta^u < \lambda^{\ast} < \delta^s < \lambda_{\ast}, \quad \quad \quad (0.1) $$

i.e. f is supercritical for |x| small and subcritical for |x| large, and

$$\label{due} f(u)= \max\{u|u|^{q^s-2}, u|u|^{q^u-2}\}, \quad 2_{\ast} < q^s <2 ^{\ast} < q^u \quad \quad \quad \quad (0.2) $$

i.e. f is subcritical for u small and supercritical for u large. We find a surprisingly rich structure for both the non-linearities, similar to the one detected by Bamon et al. for ${f=u^{q^u-1}+u^{q^s-1}}$ when 2_* < q ^s < 2^* < q ^u. More precisely if we fix q ^s and we let q ^u vary in (0.2) we find that there are no ground states for q ^u large, and an arbitrarily large number of ground states with fast decay as q ^u approaches 2^*. We also find the symmetric result when we fix q ^u and let q ^s vary. We also prove the existence of a further resonance phenomenon which generates small windows with a large number of ground states with fast decay. Similar results hold for (0.1).