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

TL;DR

This paper studies a fractional harmonic oscillator problem, establishing existence, regularity, decay properties, and nonexistence results, along with compactness of embeddings and solutions for critical and superlinear nonlinearities.

Contribution

It introduces new existence and nonexistence results, decay estimates, and compactness properties for solutions to a fractional harmonic oscillator problem.

Findings

The embedding into L^q is compact.

No non-trivial solutions for the critical power case.

Existence of solutions for superlinear critical problems.

Abstract

Under simple hypotheses on the nonlinearity $f$, we consider the fractional harmonic operator problem \begin{equation}\label{abstr}\sqrt{-\Delta+|x|^2}\,u=f(x,u)\ \ \textrm{in }\ \mathbb{R}^N\end{equation} or, since we work in the extension setting $\mathbb{R}^{N+1}_+$, $$\left\{\begin{aligned} -\Delta v +|x|^2v&=0, &&\mbox{in} \ \mathbb{R}^{N+1}_+,\\ -\displaystyle\frac{\partial v}{\partial x}(x,0)&=f(x,v(x,0)) &&\mbox{on} \ \mathbb{R}^{N}\cong\partial \mathbb{R}^{N+1}_+.\end{aligned}\right.$$ Defining the space $$\mathcal{H}(\mathbb{R}^{N+1}_+)=\left\{v\in H^1(\mathbb{R}^{N+1}_+): \iint_{\mathbb{R}^{N+1}_+}\left[|\nabla v|^2+|x|^2v^2\right]dx dy<\infty\right\},$$ 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 $f(x,u)=|u|^{p^*-2}u$ 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 $|u(x)|\leq C/|x|$ for any $|x|>M$. Finally, we prove the existence of a solution to a superlinear critical problem in the case $f(x,u)=|u|^{2^*-2}u+\lambda |u|^{q-1}$, $1<q<2^*-1$.