On a fractional harmonic oscillator: existence and inexistence of solution, regularity and decay properties

Under simple hypotheses on the nonlinearity $f$, we consider the fractional harmonic operator problem \begin{equation}\label{abstr}\sqrt{-Δ+|x|^2},u=f(x,u)\ \ $\text{in }$\ $\mathbb{R}$^N\end{equation} or, since we work in the extension setting ${\mathbb{R}}_{+}^{N+1}$, $$\{\begin{array}{rlrl}{\displaystyle -\mathrm{\Delta }v+\mathrm{\mid }x{\mathrm{\mid }}^{2}v}& {\displaystyle =0,}& {\displaystyle }& {\displaystyle in{\mathbb{R}}_{+}^{N+1},}\\ {\displaystyle -{\displaystyle \frac{\mathrm{\partial }v}{\mathrm{\partial }x}(x,0)}}& {\displaystyle =f(x,v(x,0))}& {\displaystyle }& {\displaystyle on{\mathbb{R}}^N\cong \mathrm{\partial }{\mathbb{R}}_{+}^{N+1}\mathrm{.}}\end{array}$$ Defining the space $$\mathcal{H}({\mathbb{R}}_{+}^{N+1})=\{v\in H^1({\mathbb{R}}_{+}^{N+1}):{\iint }_{{\mathbb{R}}_{+}^{N+1}}[\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2+\mathrm{\mid }x{\mathrm{\mid }}^2{v}^2]dxdy<\mathrm{\infty }\},$$ we prove that the embedding $$\mathcal{H}({\mathbb{R}}_{+}^{N+1})\hookrightarrow L^q({\mathbb{R}}^N)$$ is compact. We also obtain a Pohozaev-type identity for this problem, show that in the case $f(x,u)=\mathrm{\mid }u{\mathrm{\mid }}^{p^{\ast }-2}u$ the problem has no non-trivial solution, compare the extremal attached to this problem with the one of the space $H^1({\mathbb{R}}_{+}^{N+1})$, prove that the solution $u$ of our problem belongs to $L^p({\mathbb{R}}^N)$ for all $p\in [2,\mathrm{\infty }]$ and satisfy the polynomial decay $\mathrm{\mid }u(x)\mathrm{\mid }\le C\mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ for any $\mathrm{\mid }x\mathrm{\mid }>M$. Finally, we prove the existence of a solution to a superlinear critical problem in the case $f(x,u)=\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{q-1}$, $1<q<2^{\ast }-1$.