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

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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 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.

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