On the singularly perturbation fractional Kirchhoff equations: Critical case
This article deals with the following fractional Kirchhoff problem with critical exponent a+b∫RN∣(−Δ)s2u∣2dx(−Δ)su=(1+εK(x))u2s∗−1,inRN,\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\vareps...
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| Format: | Article |
| Language: | English |
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De Gruyter
2022-03-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2022-0234 |
| _version_ | 1811185894556696576 |
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| author | Gu Guangze Yang Zhipeng |
| author_facet | Gu Guangze Yang Zhipeng |
| author_sort | Gu Guangze |
| collection | DOAJ |
| description | This article deals with the following fractional Kirchhoff problem with critical exponent a+b∫RN∣(−Δ)s2u∣2dx(−Δ)su=(1+εK(x))u2s∗−1,inRN,\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a,b>0a,b\gt 0 are given constants, ε\varepsilon is a small parameter, 2s∗=2NN−2s{2}_{s}^{\ast }=\frac{2N}{N-2s} with 0<s<10\lt s\lt 1 and N≥4sN\ge 4s. We first prove the nondegeneracy of positive solutions when ε=0\varepsilon =0. In particular, we prove that uniqueness breaks down for dimensions N>4sN\gt 4s, i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε\varepsilon small. |
| first_indexed | 2024-04-11T13:38:11Z |
| format | Article |
| id | doaj.art-f5538267f43a45a8937abf2671b0d9b2 |
| institution | Directory Open Access Journal |
| issn | 2191-950X |
| language | English |
| last_indexed | 2024-04-11T13:38:11Z |
| publishDate | 2022-03-01 |
| publisher | De Gruyter |
| record_format | Article |
| series | Advances in Nonlinear Analysis |
| spelling | doaj.art-f5538267f43a45a8937abf2671b0d9b22022-12-22T04:21:24ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2022-03-011111097111610.1515/anona-2022-0234On the singularly perturbation fractional Kirchhoff equations: Critical caseGu Guangze0Yang Zhipeng1Department of Mathematics, Yunnan Normal University, Kunming 650500, ChinaDepartment of Mathematics, Yunnan Normal University, Kunming 650500, ChinaThis article deals with the following fractional Kirchhoff problem with critical exponent a+b∫RN∣(−Δ)s2u∣2dx(−Δ)su=(1+εK(x))u2s∗−1,inRN,\left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a,b>0a,b\gt 0 are given constants, ε\varepsilon is a small parameter, 2s∗=2NN−2s{2}_{s}^{\ast }=\frac{2N}{N-2s} with 0<s<10\lt s\lt 1 and N≥4sN\ge 4s. We first prove the nondegeneracy of positive solutions when ε=0\varepsilon =0. In particular, we prove that uniqueness breaks down for dimensions N>4sN\gt 4s, i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε\varepsilon small.https://doi.org/10.1515/anona-2022-0234fractional kirchhoff equationsnondegeneracylyapunov-schmidt reduction35a0135b2535a15 |
| spellingShingle | Gu Guangze Yang Zhipeng On the singularly perturbation fractional Kirchhoff equations: Critical case Advances in Nonlinear Analysis fractional kirchhoff equations nondegeneracy lyapunov-schmidt reduction 35a01 35b25 35a15 |
| title | On the singularly perturbation fractional Kirchhoff equations: Critical case |
| title_full | On the singularly perturbation fractional Kirchhoff equations: Critical case |
| title_fullStr | On the singularly perturbation fractional Kirchhoff equations: Critical case |
| title_full_unstemmed | On the singularly perturbation fractional Kirchhoff equations: Critical case |
| title_short | On the singularly perturbation fractional Kirchhoff equations: Critical case |
| title_sort | on the singularly perturbation fractional kirchhoff equations critical case |
| topic | fractional kirchhoff equations nondegeneracy lyapunov-schmidt reduction 35a01 35b25 35a15 |
| url | https://doi.org/10.1515/anona-2022-0234 |
| work_keys_str_mv | AT guguangze onthesingularlyperturbationfractionalkirchhoffequationscriticalcase AT yangzhipeng onthesingularlyperturbationfractionalkirchhoffequationscriticalcase |