Multiple concentrating solutions for a fractional (p, q)-Choquard equation
We focus on the following fractional (p, q)-Choquard problem: (−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=1|x|μ*F(u)f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2024-04-01
|
| Series: | Advanced Nonlinear Studies |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/ans-2023-0125 |
| _version_ | 1797197569798963200 |
|---|---|
| author | Ambrosio Vincenzo |
| author_facet | Ambrosio Vincenzo |
| author_sort | Ambrosio Vincenzo |
| collection | DOAJ |
| description | We focus on the following fractional (p, q)-Choquard problem: (−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=1|x|μ*F(u)f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN,
$\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=\left(\frac{1}{\vert x{\vert }^{\mu }}{\ast}F\left(u\right)\right)f\left(u\right) \,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \\ u\in {W}^{s,p}\left({\mathbb{R}}^{N}\right)\cap {W}^{s,q}\left({\mathbb{R}}^{N}\right), u{ >}0\,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \end{cases}$
where ɛ > 0 is a small parameter, 0 < s < 1, 1<p<q<Ns
$1{< }p{< }q{< }\frac{N}{s}$
, 0 < μ < sp, (−Δ)rs
${\left(-{\Delta}\right)}_{r}^{s}$
, with r ∈ {p, q}, is the fractional r-Laplacian operator, V:RN→R
$V:{\mathbb{R}}^{N}\to \mathbb{R}$
is a positive continuous potential satisfying a local condition, f:R→R
$f:\mathbb{R}\to \mathbb{R}$
is a continuous nonlinearity with subcritical growth at infinity and F(t)=∫0tf(τ)dτ
$F\left(t\right)={\int }_{0}^{t}f\left(\tau \right) \mathrm{d}\tau $
. Applying suitable variational and topological methods, we relate the number of solutions with the topology of the set where the potential V attains its minimum value. |
| first_indexed | 2024-04-24T06:46:03Z |
| format | Article |
| id | doaj.art-01d6d19a9628412fbffa540f7bc6f7e5 |
| institution | Directory Open Access Journal |
| issn | 2169-0375 |
| language | English |
| last_indexed | 2024-04-24T06:46:03Z |
| publishDate | 2024-04-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advanced Nonlinear Studies |
| spelling | doaj.art-01d6d19a9628412fbffa540f7bc6f7e52024-04-22T19:39:26ZengDe GruyterAdvanced Nonlinear Studies2169-03752024-04-0124251054110.1515/ans-2023-0125Multiple concentrating solutions for a fractional (p, q)-Choquard equationAmbrosio Vincenzo0Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica Delle Marche, via Brecce Bianche, 12 60131Ancona, ItalyWe focus on the following fractional (p, q)-Choquard problem: (−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=1|x|μ*F(u)f(u) in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0 in RN, $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=\left(\frac{1}{\vert x{\vert }^{\mu }}{\ast}F\left(u\right)\right)f\left(u\right) \,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \\ u\in {W}^{s,p}\left({\mathbb{R}}^{N}\right)\cap {W}^{s,q}\left({\mathbb{R}}^{N}\right), u{ >}0\,\text{in}\,{\mathbb{R}}^{N},\quad \hfill \end{cases}$ where ɛ > 0 is a small parameter, 0 < s < 1, 1<p<q<Ns $1{< }p{< }q{< }\frac{N}{s}$ , 0 < μ < sp, (−Δ)rs ${\left(-{\Delta}\right)}_{r}^{s}$ , with r ∈ {p, q}, is the fractional r-Laplacian operator, V:RN→R $V:{\mathbb{R}}^{N}\to \mathbb{R}$ is a positive continuous potential satisfying a local condition, f:R→R $f:\mathbb{R}\to \mathbb{R}$ is a continuous nonlinearity with subcritical growth at infinity and F(t)=∫0tf(τ)dτ $F\left(t\right)={\int }_{0}^{t}f\left(\tau \right) \mathrm{d}\tau $ . Applying suitable variational and topological methods, we relate the number of solutions with the topology of the set where the potential V attains its minimum value.https://doi.org/10.1515/ans-2023-0125fractional (p, q)-laplacian operatorpenalization techniqueljusternik–schnirelmann theory35a1535b3835j6035r1145k0558e05 |
| spellingShingle | Ambrosio Vincenzo Multiple concentrating solutions for a fractional (p, q)-Choquard equation Advanced Nonlinear Studies fractional (p, q)-laplacian operator penalization technique ljusternik–schnirelmann theory 35a15 35b38 35j60 35r11 45k05 58e05 |
| title | Multiple concentrating solutions for a fractional (p, q)-Choquard equation |
| title_full | Multiple concentrating solutions for a fractional (p, q)-Choquard equation |
| title_fullStr | Multiple concentrating solutions for a fractional (p, q)-Choquard equation |
| title_full_unstemmed | Multiple concentrating solutions for a fractional (p, q)-Choquard equation |
| title_short | Multiple concentrating solutions for a fractional (p, q)-Choquard equation |
| title_sort | multiple concentrating solutions for a fractional p q choquard equation |
| topic | fractional (p, q)-laplacian operator penalization technique ljusternik–schnirelmann theory 35a15 35b38 35j60 35r11 45k05 58e05 |
| url | https://doi.org/10.1515/ans-2023-0125 |
| work_keys_str_mv | AT ambrosiovincenzo multipleconcentratingsolutionsforafractionalpqchoquardequation |