Positive solutions for semilinear elliptic equations with mixed non-linearities: 2 simple models exhibiting several bifurcations.

In this paper we analyze the structure of positive radial solutions for the following semi-linear equations: $$\Delta u + f(u,|\mathbf{x}|)=0$$ where $\mathbf{x}\in \RR^n$ and $f$ is superlinear. In fact we just consider two very special non-linearities, i.e. \begin{equation}\label{uno} f(u,|\mathbf{x}|) = u|u|^{q-2} \max\{ |\mathbf{x}|^{\delta^s}, |\mathbf{x}|^{\delta^u} \} \; \quad -2<\delta^u<\lambda^*<\delta^s<\lambda_*, \end{equation} i.e. $f$ is supercritical for $|\mathbf{x}|$ small and subcritical for $|\mathbf{x}|$ large, and \begin{equation}\label{due} f(u)= \max\{ u|u|^{q^s-2}, u|u|^{q^u-2}\}, \quad 2_*<q^s<2^*<q^u \end{equation} 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 (\ref{due}) 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 (\ref{uno}).