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. egin{equation}label{uno} f(u,|mathbf{x}|) = u|u|^{q-2} max{ |mathbf{x}|^{delta^s}, |mathbf{x}|^{delta^u} } ; quad -2