Schrödinger–Kirchhoff–Hardy <inline-formula><tex-math id="M1">\begin{document} p \end{document}</tex-math></inline-formula>–fractional equations without the Ambrosetti–Rabinowitz condition

Previous Article
Regularity under general and $ p,q- $ growth conditions
DCDS-S Home
This Issue
Next Article
Remarks on mean curvature flow solitons in warped products

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 and 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.
[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

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

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.

[1]	Zedong Yang, Guotao Wang, Ravi P. Agarwal, Haiyong Xu. Existence and nonexistence of entire positive radial solutions for a class of Schrödinger elliptic systems involving a nonlinear operator. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3821-3836. doi: 10.3934/dcdss.2020436
[2]	Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254
[3]	Die Hu, Xianhua Tang, Qi Zhang. Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type. Communications on Pure and Applied Analysis, 2022, 21 (3) : 1071-1091. doi: 10.3934/cpaa.2022010
[4]	Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure and Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037
[5]	Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071
[6]	Lorenzo Brasco, Enea Parini, Marco Squassina. Stability of variational eigenvalues for the fractional $p-$Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (4) : 1813-1845. doi: 10.3934/dcds.2016.36.1813
[7]	Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[8]	Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447
[9]	John R. Graef, Lingju Kong, Min Wang. Existence of homoclinic solutions for second order difference equations with $p$-laplacian. Conference Publications, 2015, 2015 (special) : 533-539. doi: 10.3934/proc.2015.0533
[10]	Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[11]	Chungen Liu, Huabo Zhang. Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity. Communications on Pure and Applied Analysis, 2021, 20 (2) : 817-834. doi: 10.3934/cpaa.2020292
[12]	Maoding Zhen, Binlin Zhang, Xiumei Han. A new approach to get solutions for Kirchhoff-type fractional Schrödinger systems involving critical exponents. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1927-1954. doi: 10.3934/dcdsb.2021115
[13]	Yixuan Wu, Yanzhi Zhang. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 851-876. doi: 10.3934/dcdss.2022016
[14]	Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete and Continuous Dynamical Systems - S, 2021, 14 (6) : 1945-1966. doi: 10.3934/dcdss.2020469
[15]	Leijin Cao. Existence of stable standing waves for the nonlinear Schrödinger equation with the Hardy potential. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022125
[16]	Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168
[17]	Gerd Grubb. Limited regularity of solutions to fractional heat and Schrödinger equations. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3609-3634. doi: 10.3934/dcds.2019148
[18]	Kaimin Teng, Xiumei He. Ground state solutions for fractional Schrödinger equations with critical Sobolev exponent. Communications on Pure and Applied Analysis, 2016, 15 (3) : 991-1008. doi: 10.3934/cpaa.2016.15.991
[19]	Miao Du, Lixin Tian. Infinitely many solutions of the nonlinear fractional Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3407-3428. doi: 10.3934/dcdsb.2016104
[20]	Xianling Fan, Yuanzhang Zhao, Guifang Huang. Existence of solutions for the $p-$Laplacian with crossing nonlinearity. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1019-1024. doi: 10.3934/dcds.2002.8.1019

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Alessio Fiscella
on Google Scholar
Alessio Fiscella

2021 Impact Factor: 1.865

Tools

Article outline

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Alessio Fiscella
on Google Scholar
Alessio Fiscella

[Back to Top]