On elliptic problems in $\mathbf R^N$ with supercritical exponent of nonlinearity

Elliptic problems of the form
$$ \begin{cases} \Delta u+f(x,u)=h(x),\quad x\in\mathbf R^N\ \ (N\geqslant 3), \\ \displaystyle\lim_{|x|\to\infty}u(x)=0 \end{cases} $$
are considered under appropriate conditions. This class of problems includes the inhomogeneous Emden–Fowler problem
$$ \begin{cases} \Delta u+|u|^{p-2}u=h(x),\quad x\in\mathbf R^N\ \ (N\geqslant 3), \\ \displaystyle\lim_{|x|\to\infty}u(x)=0 \end{cases} $$
with $p>p_c=\dfrac{2N}{N-2}$.
The first part of this article is concerned with radial solutions, where
$$ f(x,u)=f(|x|,u) \quad\text{and}\quad h(x)=h(|x|). $$

The second part considers solvability in classes of functions with prescribed bound on decay at infinity, but without assumptions on radial symmetry.