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

Abstract

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)\ \ \textrm{in }\ \mathbb{R}^N\end{equation} or, since we work in the extension setting ${\mathbb{R}}_{+}^{N+1}$, $$\left\{\begin{aligned} -Δv +|x|^2v&=0, &&\mbox{in} \ \mathbb{R}^{N+1}_+,\\ -\displaystyle\frac{\partial v}{\partial x}(x,0)&=f(x,v(x,0)) &&\mbox{on} \ \mathbb{R}^{N}\cong\partial \mathbb{R}^{N+1}_+.\end{aligned}\right.$$ Defining the space $$\mathcal{H}({\mathbb{R}}_{+}^{N+1})=\left\{v\in H^1({\mathbb{R}}_{+}^{N+1}):{\iint }_{{\mathbb{R}}_{+}^{N+1}}\left[|\nabla v{|}^2+|x{|}^2{v}^2\right]dxdy<\infty \right\},$$ 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 $f(x,u)=|u{|}^{p^{\ast }-2}u$ 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 $p\in [2,\infty ]$ and satisfy the polynomial decay $|u(x)|\le C/|x|$ for any $|x|>M$. Finally, we prove the existence of a solution to a superlinear critical problem in the case $f(x,u)=|u{|}^{2^{\ast }-2}u+\lambda |u{|}^{q-1}$, $1<q<2^{\ast }-1$.

25 pages