Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2408.01756 |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866909279392366592 |
|---|---|
| author | Bueno, Hamilton P. Medeiros, Aldo H. S. Miyagaki, Olimpio H. Pereira, Gilberto A. |
| author_facet | Bueno, Hamilton P. Medeiros, Aldo H. S. Miyagaki, Olimpio H. Pereira, Gilberto A. |
| contents | 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 $$\mathcal{H}(\mathbb{R}^{N+1}_+)=\left\{v\in H^1(\mathbb{R}^{N+1}_+): \iint_{\mathbb{R}^{N+1}_+}\left[|\nabla v|^2+|x|^2v^2\right]dx dy<\infty\right\},$$ 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 $f(x,u)=|u|^{2^*-2}u+λ|u|^{q-1}$, $1<q<2^*-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_01756 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a fractional harmonic oscillator: existence and inexistence of solution, regularity and decay properties Bueno, Hamilton P. Medeiros, Aldo H. S. Miyagaki, Olimpio H. Pereira, Gilberto A. Analysis of PDEs 35R11, 35B33, 35B40, 35A1 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 $$\mathcal{H}(\mathbb{R}^{N+1}_+)=\left\{v\in H^1(\mathbb{R}^{N+1}_+): \iint_{\mathbb{R}^{N+1}_+}\left[|\nabla v|^2+|x|^2v^2\right]dx dy<\infty\right\},$$ 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 $f(x,u)=|u|^{2^*-2}u+λ|u|^{q-1}$, $1<q<2^*-1$. |
| title | On a fractional harmonic oscillator: existence and inexistence of solution, regularity and decay properties |
| topic | Analysis of PDEs 35R11, 35B33, 35B40, 35A1 |
| url | https://arxiv.org/abs/2408.01756 |