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July  2020, 13(7): 1993-2007. doi: 10.3934/dcdss.2020154

Schrödinger–Kirchhoff–Hardy $ p $–fractional equations without the Ambrosetti–Rabinowitz condition

Alessio Fiscella

Departamento de Matemática, Universidade Estadual de Campinas, IMECC, Rua Sérgio Buarque de Holanda, 651, Campinas, SP CEP 13083–859, Brazil

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday, with great affection and esteem

Received  May 2018 Revised  October 2018 Published  November 2019

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.
Keywords: Schrödinger–Kirchhoff equations, existence of entire solutions, fractional p–Laplacian operator, Hardy coefficients, variational methods.
Mathematics Subject Classification: Primary: 35J60, 35B08, 35R11; Secondary: 35Q55, 35B33, 47G20.
Citation: Alessio Fiscella. Schrödinger–Kirchhoff–Hardy $ p $–fractional equations without the Ambrosetti–Rabinowitz condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1993-2007. doi: 10.3934/dcdss.2020154
References:
[1]

V. Ambrosio, Multiple solutions for a fractional p-Laplacian equation with sign-changing potential, Electron. J. Differential Equations, 2016 (2016), 12 pp.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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