This paper is devoted to the study of the following Schrödinger–Kirchhoff–Hardy equation in $ \mathbb R^n $
$ M\left(\iint_{\mathbb R^{2n}}\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}dxdy\right)(-\Delta)^{s}_pu+V(x)|u|^{p-2}u-\mu\frac{|u|^{p-2}u}{|x|^{ps}} = f(x, u), $
where $ (-\Delta)^s_p $ is the fractional $ p $–Laplacian, with $ s\in(0, 1) $ and $ p>1 $, dimension $ n>ps $, $ M $ models a Kirchhoff coefficient, $ V $ is a positive potential, $ f $ is a continuous nonlinearity and $ \mu $ is a real parameter. The main feature of the paper is the combination of a Kirchhoff coefficient and a Hardy term with a suitable function $ f $ which does not necessarily satisfy the Ambrosetti–Rabinowitz condition. Under different assumptions for $ f $ and restrictions for $ \mu $, we provide existence and multiplicity results by variational methods.
Citation: |
[1] |
V. Ambrosio, Multiple solutions for a fractional p-Laplacian equation with sign-changing potential, Electron. J. Differential Equations, 2016 (2016), 12 pp.
![]() ![]() |
[2] |
G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.
doi: 10.1007/s00030-012-0193-y.![]() ![]() ![]() |
[3] |
G. Autuori, P. Pucci and C. Varga, Existence theorems for quasilinear elliptic eigenvalue problems in unbounded domains, Adv. Differential Equations, 18 (2013), 1-48.
![]() ![]() |
[4] |
T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal., 20 (1993), 1205-1216.
doi: 10.1016/0362-546X(93)90151-H.![]() ![]() ![]() |
[5] |
T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $\mathbb{R}^{N}$, Comm. Partial Differential Equations, 20 (1995), 1725-1741.
doi: 10.1080/03605309508821149.![]() ![]() ![]() |
[6] |
Z. Binlin, G. Molica Bisci and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015), 2247-2264.
doi: 10.1088/0951-7715/28/7/2247.![]() ![]() ![]() |
[7] |
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88 (1983), 486-490.
doi: 10.1090/S0002-9939-1983-0699419-3.![]() ![]() ![]() |
[8] |
M. Caponi and P. Pucci, Existence theorems for entire solutions of stationary Kirchhoff fractional p-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.
doi: 10.1007/s10231-016-0555-x.![]() ![]() ![]() |
[9] |
D. G. Costa and O. H. Miyagaki, Nontrivial solutions for perturbations of the p-Laplacian on unbounded domains, J. Math. Anal. Appl., 193 (1995), 737-755.
doi: 10.1006/jmaa.1995.1264.![]() ![]() ![]() |
[10] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Ergebnisse der Mathematik und ihrer Grenzgebiete(3), 19, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-74331-3.![]() ![]() ![]() |
[11] |
A. Fiscella and P. Pucci, p-fractional Kirchhoff equations involving critical nonlinearities, Nonlinear Anal. Real World Appl., 35 (2017), 350-378.
doi: 10.1016/j.nonrwa.2016.11.004.![]() ![]() ![]() |
[12] |
A. Fiscella and P. Pucci, Kirchhoff-Hardy fractional problems with lack of compactness, Adv. Nonlinear Stud., 17 (2017), 429-456.
doi: 10.1515/ans-2017-6021.![]() ![]() ![]() |
[13] |
A. Fiscella, P. Pucci and S. Saldi, Existence of entire solutions for Schrödinger-Hardy systems involving two fractional operators, Nonlinear Anal., 158 (2017), 109-131.
doi: 10.1016/j.na.2017.04.005.![]() ![]() ![]() |
[14] |
A. Fiscella, P. Pucci and B. L. Zhang, p-fractional Hardy-Schrödinger-Kirchhoff systems with critical nonlinearities, Adv. Nonlinear Anal., 8 (2019), 1111-1131.
doi: 10.1515/anona-2018-0033.![]() ![]() ![]() |
[15] |
V. Maz'ya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., 195 (2002), 230-238.
doi: 10.1006/jfan.2002.3955.![]() ![]() ![]() |
[16] |
X. Mingqi, V. D. Radulescu and B. L. Zhang, Combined effects for fractional Schrödinger-Kirchhoff systems with critical nonlinearities, ESAIM Control Optim. Calc. Var., 24 (2018), 1249-1273.
doi: 10.1051/cocv/2017036.![]() ![]() ![]() |
[17] |
G. Molica Bisci, D. Repovš and R. Servadei, Nontrivial solutions of superlinear nonlocal problems, Forum Math., 28 (2016), 1095-1110.
doi: 10.1515/forum-2015-0204.![]() ![]() ![]() |
[18] |
D. Mugnai and N. S. Papageorgiu, Wang's multiplicity result for superlinear (p, q)-equations without the Ambrosetti-Rabinowitz condition, Trans. Amer. Math. Soc., 366 (2014), 4919-4937.
doi: 10.1090/S0002-9947-2013-06124-7.![]() ![]() ![]() |
[19] |
P. Piersanti and P. Pucci, Entire solutions for critical p-fractional Hardy Schrödinger Kirchhoff equations, Publ. Mat., 62 (2018), 3-36.
doi: 10.5565/PUBLMAT6211801.![]() ![]() ![]() |
[20] |
P. Pucci, M. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in $\mathbb R^N$, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.
doi: 10.1007/s00526-015-0883-5.![]() ![]() ![]() |
[21] |
P. Pucci, M. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.
doi: 10.1515/anona-2015-0102.![]() ![]() ![]() |
[22] |
L. Y. Shao and H. B. Chen, Ground state of solutions for a class of fractional Schrödinger equations with critical Sobolev exponent and steep potential well, Math. Methods Appl. Sci., 40 (2017), 7255-7266.
doi: 10.1002/mma.4527.![]() ![]() ![]() |
[23] |
Y. H. Wei and X. F. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124.
doi: 10.1007/s00526-013-0706-5.![]() ![]() ![]() |
[24] |
M. Q. Xiang, B. L. Zhang and M. M. Yang, A fractional Kirchhoff-type problem in $\mathbb R^N$ without the (AR) condition, Complex Var. Elliptic Equ., 61 (2016), 1481-1493.
doi: 10.1080/17476933.2016.1182519.![]() ![]() ![]() |
[25] |
J. Zhang, Z. L. Lou, Y. J. Ji and W. Shao, Ground state of Kirchhoff type fractional Schrödinger equations with critical growth, J. Math. Anal. Appl., 462 (2018), 57-83.
doi: 10.1016/j.jmaa.2018.01.060.![]() ![]() ![]() |
[26] |
Y. P. Zhang, X. H. Tang and J. Zhang, Existence of infinitely many solutions for fractional p-Laplacian with sign-changing potential, Electron. J. Differential Equations, 2017 (2017), 14 pp.
![]() ![]() |