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

Abstract

Supposing only that $\underset{t\to 0}{\lim }\frac{f(t)}{t}=0$ and $\underset{t\to \infty }{\lim }\frac{f(t)}{t^p}=0$, 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} -Δu+ 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} -Δu +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-α}}*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.

23 pages. arXiv admin note: text overlap with arXiv:1802.03963