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.
Categories: math.AP