Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities
In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0,...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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De Gruyter
2024-03-01
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| Series: | Advanced Nonlinear Studies |
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| Online Access: | https://doi.org/10.1515/ans-2023-0110 |
| _version_ | 1827276372248100864 |
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| author | Cingolani Silvia Gallo Marco Tanaka Kazunaga |
| author_facet | Cingolani Silvia Gallo Marco Tanaka Kazunaga |
| author_sort | Cingolani Silvia |
| collection | DOAJ |
| description | In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN,
${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$
(*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α
${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$
is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN)
$u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$
, by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0
${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$
prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C
1-regularity. |
| first_indexed | 2024-04-24T06:46:29Z |
| format | Article |
| id | doaj.art-45c1dc70ce1b4a2b9fad14a6bea3f000 |
| institution | Directory Open Access Journal |
| issn | 2169-0375 |
| language | English |
| last_indexed | 2024-04-24T06:46:29Z |
| publishDate | 2024-03-01 |
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| series | Advanced Nonlinear Studies |
| spelling | doaj.art-45c1dc70ce1b4a2b9fad14a6bea3f0002024-04-22T19:39:26ZengDe GruyterAdvanced Nonlinear Studies2169-03752024-03-0124230333410.1515/ans-2023-0110Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identitiesCingolani Silvia0Gallo Marco1Tanaka Kazunaga2Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125Bari, ItalyDipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via della Garzetta 48, 25133Brescia, ItalyDepartment of Mathematics, School of Science and Engineering, Waseda University, 3-4-1 Ohkubo, Tokyo169-8555, JapanIn this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation (−Δ)su+μu=(Iα*F(u))F′(u) inRN, ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), Iα∼1|x|N−α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert }^{N-\alpha }}$ is the Riesz potential, and F is a general subcritical nonlinearity. The goal is to prove existence of multiple (radially symmetric) solutions u∈Hs(RN) $u\in {H}^{s}\left({\mathbb{R}}^{N}\right)$ , by assuming F odd or even: we consider both the case μ > 0 fixed and the case ∫RNu2=m>0 ${\int }_{{\mathbb{R}}^{N}}{u}^{2}=m{ >}0$ prescribed. Here we also simplify some arguments developed for s = 1 (S. Cingolani, M. Gallo, and K. Tanaka, “Multiple solutions for the nonlinear Choquard equation with even or odd nonlinearities,” Calc. Var. Partial Differ. Equ., vol. 61, no. 68, p. 34, 2022). A key point in the proof is given by the research of suitable multidimensional odd paths, which was done in the local case by Berestycki and Lions (H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations II: existence of infinitely many solutions,” Arch. Ration. Mech. Anal., vol. 82, no. 4, pp. 347–375, 1983); for (*) the nonlocalities play indeed a special role. In particular, some properties of these paths are needed in the asymptotic study (as μ varies) of the mountain pass values of the unconstrained problem, then exploited to describe the geometry of the constrained problem and detect infinitely many normalized solutions for any m > 0. The found solutions satisfy in addition a Pohozaev identity: in this paper we further investigate the validity of this identity for solutions of doubly nonlocal equations under a C 1-regularity.https://doi.org/10.1515/ans-2023-0110fractional laplaciannonlinear choquard pekar equationdouble nonlocalitynormalized solutionsinfinitely many solutionspohozaev identity35a0135a1535b0635b3835d3035j1535j2035j6135q4035q5535q6035q7035q7535q8535q9235r0935r1145k0547g1047j3049j3558e0558j05 |
| spellingShingle | Cingolani Silvia Gallo Marco Tanaka Kazunaga Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities Advanced Nonlinear Studies fractional laplacian nonlinear choquard pekar equation double nonlocality normalized solutions infinitely many solutions pohozaev identity 35a01 35a15 35b06 35b38 35d30 35j15 35j20 35j61 35q40 35q55 35q60 35q70 35q75 35q85 35q92 35r09 35r11 45k05 47g10 47j30 49j35 58e05 58j05 |
| title | Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities |
| title_full | Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities |
| title_fullStr | Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities |
| title_full_unstemmed | Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities |
| title_short | Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities |
| title_sort | infinitely many free or prescribed mass solutions for fractional hartree equations and pohozaev identities |
| topic | fractional laplacian nonlinear choquard pekar equation double nonlocality normalized solutions infinitely many solutions pohozaev identity 35a01 35a15 35b06 35b38 35d30 35j15 35j20 35j61 35q40 35q55 35q60 35q70 35q75 35q85 35q92 35r09 35r11 45k05 47g10 47j30 49j35 58e05 58j05 |
| url | https://doi.org/10.1515/ans-2023-0110 |
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