Multiple solutions for a fractional Laplacian system involving critical Sobolev-Hardy exponents and homogeneous term
Abstract
In this paper, we deal with a class of fractional Laplacian system with critical Sobolev-Hardy exponents and sign-changing weight functions in a bounded domain. By exploiting the Nehari manifold and variational methods, some new existence and multiplicity results are obtain.
Keyword : fractional Laplacian system, Nehari manifold, critical Sobolev-Hardy exponent, homogeneous term
How to Cite
Zhang, J., & Hsu, T.-S. (2020). Multiple solutions for a fractional Laplacian system involving critical Sobolev-Hardy exponents and homogeneous term. Mathematical Modelling and Analysis, 25(1), 1-20. https://doi.org/10.3846/mma.2020.7704
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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T.-S. Hsu, H.-L. Lin and C.-C Hu. Multiple positive solutions of quasilinear elliptic equations in RN. J. Math. Anal. Appl., 388:500–512, 2012.
T. Mukherjee and K. Sreenadh. On doubly nonlocal p-fractional coupled elliptic system. Topological Meth. Nonl. Anal., 51(2):609–636, 2018. https://doi.org/10.12775/TMNA.2018.018
E. Di Nezza, G. Palatucci and E. Valdinoci. Hitchhikers guide to the fractional Sobolev spaces. Bull Sci. Math., 136:521–573, 2012. https://doi.org/10.1016/j.bulsci.2011.12.004
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S. Shakerian. Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave-convex nonlinearities. Cornell University, 2017. Available from Internet: https://arxiv.org/abs/1708.01369
L. Silvestre. Regularity of the obstacle problem for a fractional power of the Laplace operator. Com. Pure Appl. Math., 60:67–112, 2007. https://doi.org/10.1002/cpa.20153
J. Zhang and T.-S. Hsu. Nonlocal elliptic systems involving critical SobolevHardy exponents and concave-convex nonlinearities. Taiwanese J. Math., 2019. https://doi.org/10.11650/tjm/190109
J. Zhang and X. Liu. Three solutions for a fractional elliptic problems with critical and supercritical growth. Acta Math. Sci., 36B(6):1–13, 2016. https://doi.org/10.1016/S0252-9602(16)30108-4
J. Zhang, X. Liu and H. Jiao. Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity. Topological Meth. Nonl. Anal., 53(1):151–182, 2019. https://doi.org/10.12775/TMNA.2018.043
X. Cabre and Y. Sire. Nonlinear equations for fractional Laplacians, I: regularity, maximum principles, and Hamiltonian estimates. Ann Inst H Poincaré Anal Non Linéaire, 31:23–53, 2014. https://doi.org/10.1016/j.anihpc.2013.02.001
L. Cai, J. Liang and J. Zhang. Properties of solutions for a coupled fractional nonlinear system. J. Nonlinear and Convex Anal., 19:323–344, 2018.
D.C. de M. Filho and M.A.S. Souto. Systems of p-Laplacian equations involving homogenous nonlinearities with critical Sobolev exponent degrees. Comm. Part. Diff. Equ., 24:1537–1553, 1999. https://doi.org/10.1080/03605309908821473
P. Drabek and S.I. Pohozaev. Positive solutions for the p-Laplacian: application of the fibering method. Proc. Roy. Soc. Edinburh Sect. A, 127:703–727, 1997. https://doi.org/10.1017/S0308210500023787
N. Ghoussoub, F. Robert, S. Shakerian and M. Zhao. Mass and asymptotics associated to fractional Hardy-Schro¨dinger operators in critical regimes. Comm. Part. Diff. Equ., 43(6):859–892, 2018. https://doi.org/10.1080/03605302.2018.1476528
N. Ghoussoub and S. Shakerian. Borderline variational problems involving fractional Laplacians and critical singularities. Adv. Nonl. Studies, 15:527–555, 2015. https://doi.org/10.1515/ans-2015-0302
J. Giacomoni, T. Mukherjee and K. Sreenadh. Doubly nonlocal system with Hardy-Littlewood-Sobolev critical nonlinearity. J. Math. Anal. Appl., 467:638– 672, 2018. https://doi.org/10.1016/j.jmaa.2018.07.035
S. Goyal and K. Sreenadh. Existence and multiplicity of solutions for p-fractional Laplace equation with sign-changing nonlinearities. Adv. Nonl. Anal., 4(1):37– 58, 2015. https://doi.org/10.1515/anona-2014-0017
S. Goyal and K. Sreenadh. Nehari manifold for non-local elliptic operator with concave-convex nonlinearities and sign-changing weight functions. Proceedings of Indian Academy of Sci., 125(6):545–558, 2015. https://doi.org/10.1007/s12044015-0244-5
T.-S. Hsu. Multiple positive solutions for a quasilinear elliptic problem involving critical Sobolev-Hardy exponents and concave-convex nonlinearities. Nonlinear Anal., 74:3934–3944, 2011. https://doi.org/10.1016/j.na.2011.02.036
T.-S. Hsu and H.-L. Lin. Multiple positive solutions for singular elliptic equations with weighted Hardy terms and critical Sobolev-Hardy exponents. Proc. Roy. Soc. Edinburgh Sect. A., 140:617–633, 2010. https://doi.org/10.1017/S0308210509000729
T.-S. Hsu and H.-L. Lin. Multiplicity of positive solutions for semilinear elliptic systems with critical Sobolev-Hardy and Concave exponents. Acta. Math. Sci., 31(B):791–804, 2011. https://doi.org/10.1016/S0252-9602(11)60276-2
T.-S. Hsu, H.-L. Lin and C.-C Hu. Multiple positive solutions of quasilinear elliptic equations in RN. J. Math. Anal. Appl., 388:500–512, 2012.
T. Mukherjee and K. Sreenadh. On doubly nonlocal p-fractional coupled elliptic system. Topological Meth. Nonl. Anal., 51(2):609–636, 2018. https://doi.org/10.12775/TMNA.2018.018
E. Di Nezza, G. Palatucci and E. Valdinoci. Hitchhikers guide to the fractional Sobolev spaces. Bull Sci. Math., 136:521–573, 2012. https://doi.org/10.1016/j.bulsci.2011.12.004
X. Ros-Oton and J. Serra. The Pohozaev identity for the fractional Laplacian. Arch. Rational Mech. Anal., 213:587–628, 2014. https://doi.org/10.1007/s00205-014-0740-2
S. Shakerian. Multiple positive solutions for nonlocal elliptic problems involving the Hardy potential and concave-convex nonlinearities. Cornell University, 2017. Available from Internet: https://arxiv.org/abs/1708.01369
L. Silvestre. Regularity of the obstacle problem for a fractional power of the Laplace operator. Com. Pure Appl. Math., 60:67–112, 2007. https://doi.org/10.1002/cpa.20153
J. Zhang and T.-S. Hsu. Nonlocal elliptic systems involving critical SobolevHardy exponents and concave-convex nonlinearities. Taiwanese J. Math., 2019. https://doi.org/10.11650/tjm/190109
J. Zhang and X. Liu. Three solutions for a fractional elliptic problems with critical and supercritical growth. Acta Math. Sci., 36B(6):1–13, 2016. https://doi.org/10.1016/S0252-9602(16)30108-4
J. Zhang, X. Liu and H. Jiao. Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity. Topological Meth. Nonl. Anal., 53(1):151–182, 2019. https://doi.org/10.12775/TMNA.2018.043