Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent
Abstract In this paper, we study the following critical system with fractional Laplacian: {(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω, $$\textstyle\begin{cases} (-\Delta)^{s}u+\lambda_{1}u=\mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha \gamma}{2^{\ast}}|u...
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doaj-2e33578753c54d0580ab51800baaeb8e2020-11-24T20:59:02ZengSpringerOpenBoundary Value Problems1687-27702018-06-012018112510.1186/s13661-018-1016-9Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponentMaoding Zhen0Jinchun He1Haoyuan Xu2Meihua Yang3School of Mathematics and Statistics, Huazhong University of Science and TechnologySchool of Mathematics and Statistics, Huazhong University of Science and TechnologySchool of Mathematics and Statistics, Huazhong University of Science and TechnologySchool of Mathematics and Statistics, Huazhong University of Science and TechnologyAbstract In this paper, we study the following critical system with fractional Laplacian: {(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω, $$\textstyle\begin{cases} (-\Delta)^{s}u+\lambda_{1}u=\mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha \gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} & \text{in } \Omega, \\ (-\Delta)^{s}v+\lambda_{2}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta \gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v & \text{in } \Omega, \\ u=v=0 & \text{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} $$ where (−Δ)s $(-\Delta)^{s}$ is the fractional Laplacian, 0<s<1 $0< s<1$, μ1,μ2>0 $\mu_{1},\mu_{2}>0$, 2∗=2NN−2s $2^{\ast}=\frac{2N}{N-2s}$ is a fractional critical Sobolev exponent, N>2s $N>2s$, 1<α $1<\alpha$, β<2 $\beta<2$, α+β=2∗ $\alpha+\beta=2^{\ast}$, Ω is an open bounded set of RN $\mathbb{R}^{N}$ with Lipschitz boundary and λ1,λ2>−λ1,s(Ω) $\lambda_{1},\lambda_{2}>-\lambda_{1,s}(\Omega)$, λ1,s(Ω) $\lambda_{1,s}(\Omega)$ is the first eigenvalue of the non-local operator (−Δ)s $(-\Delta)^{s}$ with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all γ>0 $\gamma>0$. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when γ→0 $\gamma\rightarrow0$.http://link.springer.com/article/10.1186/s13661-018-1016-9Fractional LaplacianCritical exponentGround state solutionHigher energy solution |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Maoding Zhen Jinchun He Haoyuan Xu Meihua Yang |
spellingShingle |
Maoding Zhen Jinchun He Haoyuan Xu Meihua Yang Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent Boundary Value Problems Fractional Laplacian Critical exponent Ground state solution Higher energy solution |
author_facet |
Maoding Zhen Jinchun He Haoyuan Xu Meihua Yang |
author_sort |
Maoding Zhen |
title |
Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent |
title_short |
Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent |
title_full |
Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent |
title_fullStr |
Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent |
title_full_unstemmed |
Multiple positive solutions for nonlinear coupled fractional Laplacian system with critical exponent |
title_sort |
multiple positive solutions for nonlinear coupled fractional laplacian system with critical exponent |
publisher |
SpringerOpen |
series |
Boundary Value Problems |
issn |
1687-2770 |
publishDate |
2018-06-01 |
description |
Abstract In this paper, we study the following critical system with fractional Laplacian: {(−Δ)su+λ1u=μ1|u|2∗−2u+αγ2∗|u|α−2u|v|βin Ω,(−Δ)sv+λ2v=μ2|v|2∗−2v+βγ2∗|u|α|v|β−2vin Ω,u=v=0in RN∖Ω, $$\textstyle\begin{cases} (-\Delta)^{s}u+\lambda_{1}u=\mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha \gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} & \text{in } \Omega, \\ (-\Delta)^{s}v+\lambda_{2}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta \gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v & \text{in } \Omega, \\ u=v=0 & \text{in } \mathbb{R}^{N}\setminus\Omega, \end{cases} $$ where (−Δ)s $(-\Delta)^{s}$ is the fractional Laplacian, 0<s<1 $0< s<1$, μ1,μ2>0 $\mu_{1},\mu_{2}>0$, 2∗=2NN−2s $2^{\ast}=\frac{2N}{N-2s}$ is a fractional critical Sobolev exponent, N>2s $N>2s$, 1<α $1<\alpha$, β<2 $\beta<2$, α+β=2∗ $\alpha+\beta=2^{\ast}$, Ω is an open bounded set of RN $\mathbb{R}^{N}$ with Lipschitz boundary and λ1,λ2>−λ1,s(Ω) $\lambda_{1},\lambda_{2}>-\lambda_{1,s}(\Omega)$, λ1,s(Ω) $\lambda_{1,s}(\Omega)$ is the first eigenvalue of the non-local operator (−Δ)s $(-\Delta)^{s}$ with homogeneous Dirichlet boundary datum. By using the Nehari manifold, we prove the existence of a positive ground state solution of the system for all γ>0 $\gamma>0$. Via a perturbation argument and using the topological degree and a pseudo-gradient vector field, we show that this system has a positive higher energy solution. Then the asymptotic behaviors of the positive ground state solutions are analyzed when γ→0 $\gamma\rightarrow0$. |
topic |
Fractional Laplacian Critical exponent Ground state solution Higher energy solution |
url |
http://link.springer.com/article/10.1186/s13661-018-1016-9 |
work_keys_str_mv |
AT maodingzhen multiplepositivesolutionsfornonlinearcoupledfractionallaplaciansystemwithcriticalexponent AT jinchunhe multiplepositivesolutionsfornonlinearcoupledfractionallaplaciansystemwithcriticalexponent AT haoyuanxu multiplepositivesolutionsfornonlinearcoupledfractionallaplaciansystemwithcriticalexponent AT meihuayang multiplepositivesolutionsfornonlinearcoupledfractionallaplaciansystemwithcriticalexponent |
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1716784031380537344 |