Compact embedding from variable-order Sobolev space to $L^{q(x)}(Ω)$ and its application to Choquard equation with variable order and variable critical exponent
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866909431111876608 |
|---|---|
| author | Sakuma, Masaki |
| author_facet | Sakuma, Masaki |
| contents | In this paper, we prove the compact embedding from the variable-order Sobolev space $W^{s(x,y),p(x,y)}_0 (Ω)$ to the Nakano space $L^{q(x)}(Ω)$ with a critical exponent $q(x)$ satisfying some conditions. It is noteworthy that the embedding can be compact even when $q(x)$ reaches the critical Sobolev exponent $p_s^*(x)$. As an application, we obtain a nontrivial solution of the Choquard equation \begin{equation*} \displaystyle (-Δ)_{p(\cdot,\cdot)}^{s(\cdot,\cdot)}u+|u|^{p(x,x)-2}u=\left(\int_Ω\frac{|u(y)|^{r(y)}}{|x-y|^{\frac{α(x)+α(y)}{2}}}dy\right) |u(x)|^{r(x)-2}u(x)\quad\text{in $Ω$} \end{equation*} with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04602 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Compact embedding from variable-order Sobolev space to $L^{q(x)}(Ω)$ and its application to Choquard equation with variable order and variable critical exponent Sakuma, Masaki Analysis of PDEs 35J92, 35A15, 35B33, 35R11, 35A01 In this paper, we prove the compact embedding from the variable-order Sobolev space $W^{s(x,y),p(x,y)}_0 (Ω)$ to the Nakano space $L^{q(x)}(Ω)$ with a critical exponent $q(x)$ satisfying some conditions. It is noteworthy that the embedding can be compact even when $q(x)$ reaches the critical Sobolev exponent $p_s^*(x)$. As an application, we obtain a nontrivial solution of the Choquard equation \begin{equation*} \displaystyle (-Δ)_{p(\cdot,\cdot)}^{s(\cdot,\cdot)}u+|u|^{p(x,x)-2}u=\left(\int_Ω\frac{|u(y)|^{r(y)}}{|x-y|^{\frac{α(x)+α(y)}{2}}}dy\right) |u(x)|^{r(x)-2}u(x)\quad\text{in $Ω$} \end{equation*} with variable upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality under an appropriate boundary condition. |
| title | Compact embedding from variable-order Sobolev space to $L^{q(x)}(Ω)$ and its application to Choquard equation with variable order and variable critical exponent |
| topic | Analysis of PDEs 35J92, 35A15, 35B33, 35R11, 35A01 |
| url | https://arxiv.org/abs/2408.04602 |