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* < qs < 2* < qu. More precisely if we fix qs and we let qu vary in (0.2) we find that there are no ground states for qu large, and an arbitrarily large number of ground states with fast decay as qu approaches 2*. We also find the symmetric result when we fix qu and let qs 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).