On doubly nonlocal $p$-fractional coupled elliptic system

Tuhina Mukherjee, Konijeti Sreenadh

DOI: http://dx.doi.org/10.12775/TMNA.2018.018


We study the following nonlinear system with perturbations involving $p$-fractional Laplacian: \begin{equation} \begin{cases} (-\Delta)^s_p u+ a_1(x)u|u|^{p-2} = \alpha(|x|^{-\mu}*|u|^q)|u|^{q-2}u\\ \hskip 2.5 cm + \beta (|x|^{-\mu}*|v|^q)|u|^{q-2}u+ f_1(x)&\text{in } \mathbb R^n, \\ (-\Delta)^s_p v+ a_2(x)v|v|^{p-2} = \gamma(|x|^{-\mu}*|v|^q)|v|^{q-2}v \\ \hskip 2.5cm + \beta (|x|^{-\mu}*|u|^q)|v|^{q-2}v+ f_2(x)& \text{in } \mathbb R^n, \end{cases} \tag{P} \end{equation} where $n> sp$, $0< s< 1$, $p\geq2$, $\mu \in (0,n)$, ${p}( 2-{\mu}/{n})/2 < q < {p^*_s}( 2-{\mu}/{n})/2$, $\alpha,\beta,\gamma > 0$, $0< a_i \in C(\mathbb R^n, \mathbb R)$, $i=1,2$ and $f_1,f_2\colon \mathbb R^n \to \mathbb R$ are perturbations. We show existence of at least two nontrivial solutions for (P) using Nehari manifold and minimax methods.


p-fractional Laplacian; Choquard equation; Nehari manifold

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Adimurthi, J. Giacomoni and S. Santra, Positive solutions to a fractional equation with singular nonlinearity, arXiv:1706.01965.

C.O. Alves, M.G. Figueiredo and M. Yang, Existence of solutions for a nonlinear Choquard equation with potential vanishing at infinity, Adv. Nonlinear Anal. 5 (2016), 331–345.

C.O. Alves, A.B. Nóbrega and M. Yang, Multi-bump solutions for Choquard equation with deepening potential well, Calc. Var. (2016), 55:48.

C.O. Alves and M. Yang, Existence of semiclassical ground state solutions for a generalized Choquard equation, J. Differential Equations 257 (2014), 4133–164.

C.O. Alves and M. Yang, Multiplicity and concentration of solutions for a quasilinear Choquard equation, J. Math. Phys. 55 (2014), no. 6, 061502, 21 pp.

C.O. Alves and M. Yang, Investigating the multiplicity and concentration behaviour of solutions for a quasi-linear Choquard equation via the penalization method, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), 23–58.

L. Brasco and E. Parini, The second eigenvalue of the fractional p-Laplacian, Adv. Calc. Var. 9 (2015), 323–355.

L. Brasco, E. Parini and M. Squassina, Stability of variational eigenvalues for thefractional p-Laplacian, Discrete Contin. Dyn. Syst. Ser. A 36 (2016), 1813–1845.

L.A. Caffarelli, Nonlocal equations, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia 7 (2012), 37–52.

K.C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer, Berlin, 2005.

W. Chen and S. Deng, The Nehari manifold for a fractional p-Laplacian system involving concave-convex nonlinearities, Nonlinear Anal. Real World Appl. 27 (2016), 80–92.

W. Chen and M. Squassina, Critical nonlocal systems with concave-convex powers, Adv. Nonlinear Stud. 16 (2016), 821–842.

W. Choi, On strongly indefinite systems involving the fractional Laplacian, Nonlinear Anal. 120 (2015), 127–153.

M. Clapp and D. Salazar, Positive and sign changing solutions to a nonlinear Choquard equation, J. Math. Anal. Appl. 407 (2013), 1–15.

V. Coti Zelati and P.H. Rabinowitz, Homoclinic type solutions for a semilinear elliptic PDE on Rn , Comm. Pure Appl. Math. 45 (1992), 1217–1269.

P. d’Avenia, G. Siciliano and M. Squassina, On fractional Choquard equation, Math. Models Methods Appl. Sci. 25 (2015), 1447–1476.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.

L.F.O. Faria, O.H. Miyagaki, F.R. Pereira, M. Squassina and C. Zhang, The Brezis–Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal. 5 (2016), 85–103.

J. Fröhlich, T.-P. Tasi and H.-T. Yau, On a classical limit of quantum theory and the non-linear Hartree eqution, Visions in Mathematics. Modern Birkhäuser Classics (N. Alon, J. Bourgain, A. Connes, M. Gromov, V. Milman, eds.), Birkhäuser, Basel, 2010.

J. Fröhlich, T.-P. Tasi and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Commun. Math. Phys. 225 (2002), 223–274.

B. Ge, Multiple solutions of nonlinear Schrödinger equation with the fractional Laplacian, Nonlinear Anal. Real World Appl. 30 (2016), 236–247.

M. Ghimenti and J.V. Schaftingen, Nodal solutions for the Choquard equation, J. Funct. Anal. 271 (2016), 107–135.

M. Ghimenti, V. Moroz and J.V. Schaftingen, Least action nodal solutions for the quadratic Choquard equation, Proc. Amer. Math. Soc. 145 (2017), 737–747.

J. Giacomoni, P.K. Mishra and K. Sreenadh, Critical growth fractional elliptic systems with exponential nonlinearity, Nonlinear Anal. 136 (2016), 117–135.

J. Giacomoni, P.K. Mishra and K. Sreenadh, Fractional elliptic systems with exponential nonlinearity, Nonlinear Anal. 136 (2016), 117–135.

S. Goyal, Multiplicity results of fractional p-Laplace equations with sign-changing and singular nonlinearity, Complex Var. Elliptic Equ. 62 (2017), 158–183.

S. Goyal and K. Sreenadh, Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function, Adv. Nonlinear Anal. 4 (2015), 37–58.

Z. Guo, S. Luo and W. Zou, On critical systems involving frcational Laplacian, J. Math. Anal. Appl. 446 (2017), 681–706.

X. He, M. Squassina and W. Zou, The Nehari manifold for fractional systems involving critical nonlinearities, Comm. Pure Appl. Math. 15 (2016), 1285–1308.

A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional p-Laplacian problems via Morse theory, Adv. Calc. Var. 9 (2014), no. 2, 101–125.

A. Iannizzotto, S. Mosconi and M. Squassina, Global Hölder regularity for the fractional p-Laplacian, Rev. Mat. Iberoam. 32 (2016), 1353–1392.

A. Iannizzotto and M. Squassina, Weyl-type laws for fractional p-eigenvalue problems, Asymptot. Anal. 88 (2014), no. 4, 233–245.

E. Lieb and M. Loss, Analysis, Graduate Studies in Mathematics, Amer. Math. Soc., Providence, Rhode Island, 2001.

E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. (2014), 49:795.

D. Lü and G. Xu, On nonlinear fractional Schrödinger equations with Hartree-type nonlinearity, Appl. Anal. (2016), DOI:10.1080/00036811.2016.1260708.

G. Molica Bisci and V. Rădulescu and R. Servadei, Variational methods for nonlocal fractional problems, with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, Vol. 162, Cambridge University Press, Cambridge, 2016.

V. Moroz and J.V. Schaftingen, Groundstates of nonlinear Choquard equations: Existence, qualitative properties and decay asymptotics, J. Func. Anal. 265 (2013), no. 2, 153–184.

V. Moroz and J.V. Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015), 6557–6579.

V. Moroz and J.V. Schaftingen, Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent, Commun. Contemp. Math. 17 (2015), no. 5, 1550005, 12 pp.

V. Moroz and J.V. Schaftingen, Semi-classical states for the Choquard equation, Calc. Var. 52 (2015), 199–235.

T. Mukherjee and K. Sreenadh, Fractional Choquard equation with critical nonlinearities, Nonlinear Differential Equations and Applications 24 (2017), Art. 63.

R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal. 2 (2013), no. 3, 235–270.

R. Servadei, A critical fractional Laplace equation in the resonant case, Topol. Methods Nonlinear Anal. 43 (2014), 251–267.

R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102.

Z. Shen, F. Gao and M. Yang, Ground states for nonlinear fractional Choquard equations with general nonlinearities, Math. Methods Appl. Sci. 39 (2016), DOI: 10.1002/mma.3849.

E. Stein, Singular integrals and differentiability properties of functions, Princeton, NJ, Princeton University Press, (1970).

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 281–304.


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