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Schrödinger–Kirchhoff–Hardy $ p $–fractional equations without the Ambrosetti–Rabinowitz condition
Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda, 651, Campinas, SP CEP 13083–859, Brazil |
$ \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), $ |
$ (-\Delta)^s_p $ |
$ p $ |
$ s\in(0, 1) $ |
$ p>1 $ |
$ n>ps $ |
$ M $ |
$ V $ |
$ f $ |
$ \mu $ |
$ f $ |
$ f $ |
$ \mu $ |
References:
[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. |
show all references
References:
[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. |
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