Existence, regularity, asymptotic decay and radiality of solutions to some extension problems

Supposing only that \(\displaystyle\lim_{t \to 0} \frac{f(t)}{t} = 0\) and \(\displaystyle\lim_{t \to \infty} \frac{f(t)}{t^{p}} = 0\), for some \(p \in \left(1,\frac{N+1}{N-1}\right)\), we prove that solutions to the extension problem \begin{equation*}\left\{ \begin{array}{rcll} -\Delta u+ m^2u &=& 0, &\mbox{in} \ \ \mathbb{R}^{N+1}_{+} \\ -\frac{\partial u}{\partial{x}} (0,y)& =& f(u(0,y)), & y \in \mathbb{R}^{N}, \end{array}\right. \end{equation*} and also to the extension Hartree problem \begin{equation*} \left\{\begin{aligned} -\Delta u +m^2u&=0, &&\mbox{in} \ \mathbb{R}^{N+1}_+,\\ -\displaystyle\frac{\partial u}{\partial x}(0,y)&=-V_\infty u(0,y)+\left(\frac{1}{|y|^{N-\alpha}}*F(u(0,y))\right)f(u(0,y)) &&\mbox{in} \ \mathbb{R}^{N}\end{aligned}\right. \end{equation*} are radially symmetric in \(\mathbb{R}^N\). In the last problem, \(V_\infty>0\) is a constant and \(F\) the primitive of \(f\). Under the same hypotheses, regularity and exponential decay of solutions to the first problem is also proved and, supposing the traditional Ambrosetti-Rabinowitz condition, also existence of a ground state solution.