The nontrivial solutions for nonlinear fractional Schrödinger-Poisson system involving new fractional operator

Year 2023, Volume: 7 Issue: 1, 121 - 132, 31.03.2023

Boutebba Hamza , Hakim Lakhal , Slimani Kamel , Belhadi Tahar

https://doi.org/10.31197/atnaa.1141136

Abstract

In this paper, we investigate the existence of nontrivial solutions in the Bessel Potential space for nonlinear
fractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using the
mountain pass theorem in combination with the perturbation method, we prove the existence of solutions.

Keywords

Fractional Schrödinger-Poisson system, Mountain Pass Theorem, Bessel potential space, perturbation method

References

• [1] A. Azzollini, P. Alessio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008) 90-108.
• [2] A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Diff. Equa., 249 (2010) 1746-1763.
• [3] A.M. Batista, M.F. Furtado, Positive and nodal solutions for a nonlinear Schrödinger-Poisson system with sign-changing potentials, Nonlinear Anal. Real World Appl., 39 (2018) 142-156.
• [4] V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topological Methods Nonlinear Anal., 11 (1998) 283-293.
• [5] Z.Binlin, G.M.Bisci, R.Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015) 2247.
• [6] C. Bucur, E. Valdinoci, Nonlocal di?.appl. Cham Springer, 20 (2016).
• [7] G. Che, H. Chen, Multiplicity and concentration of solutions for a fractional Schrödinger-Poisson system with sign-changing potential, Appl. Anal., (2021) 1-22.
• [8] J. Chen, X. Tang, H. Luo, Infinitely many solutions for fractional Schrödinger-Poisson systems with sign-changing potential, Elec. J. Di?. Equa., 97 (2017).
• [9] T. D'Aprile, D. Mugnai, Solitary waves for nonlinear Klein Gordon Maxwell and Schrödinger-Maxwell equations, Proc. Royal Soc. Edinburgh. Sec. A Math., 134 (2004) 893-906.
• [10] E. Di Nezza, G. Palatucci, E.Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. scie. math., 136 (2012) 521-573.
• [11] P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Royal Soc. Edinburgh Sec. A Math., 142 (2012) 1237-1262.
• [12] X. He, W. Zou, Multiplicity of concentrating positive solutions for Schrödinger-Poisson equations with critical growth, Nonlinear Anal., 170 (2018) 142-170.
• [13] R. Jiang, C. Zhai., Two nontrivial solutions for a nonhomogeneous fractional Schrödinger-Poisson equation in R 3 , Boun. Val. Prob., 1 (2020) 1-18.
• [14] T. Jin, Multiplicity of solutions for a fractional Schrödinger-Poisson system without (PS) condition, AIMS Math., 6 (2021) 9048-9058.
• [15] K. Li, Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations, Appl. Math. Lett., 72 (2017) 1-9.
• [16] C.W. Lo, J.F. Rodrigues, On a class of fractional obstacle type problems related to the distributional Riesz derivative, arXiv prep. arXiv, 2101.06863 (2021).
• [17] Y. Meng, X. Zhang, X. He, Ground state solutions for a class of fractional Schrödinger-Poisson system with critical growth and vanishing potentials, Advances in Nonlinear Anal., 10 (2021) 1328-1355.
• [18] E.G. Murcia, G. Siciliano, Least energy radial sign-changing solution for the Schrödinger-Poisson system in R 3 under an asymptotically cubic nonlinearity, J. Math. Anal. Appl., 474 (2019) 544-571.
• [19] Q.Y. Peng, Z.Q. Ou, Y. Lv, Ground state solutions for the fractional Schrödinger-Poisson system with critical growth, Chaos, Solitons and Frac., 144 (2021) 110650.
• [20] L. Shen, Existence result for fractional Schrödinger-Poisson systems involving a Bessel operator without Ambrosetti- Rabinowitz condition, Computers and Math. Appl., 75 (2018) 296-306.
• [21] T.T. Shieh, D.E. Spector, On a new class of fractional partial differential equations, Advances in Calc. Var., 8 (2015) 321-336.
• [22] T.T. Shieh, D.E. Spector, On a new class of fractional partial di?erential equations II." Advances in Calc. Var., 11 (2018) 289-307.
• [23] M. Silhavý, Fractional vector analysis based on invariance requirements (critique of coordinate approaches), Continuum Mech.Thermodynamics, 32 (2020) 207-228.
• [24] E. M.Stein, Singular Integrals and Differentiability Properties of Functions (PMS-30), 30. Princeton university press, (2016).
• [25] K. Teng, Existence of ground state solutions for the nonlinear fractional Schrödinger-Poisson system with critical Sobolev exponent, J. Diff. Equa., 261 (2016) 3061-3106.