The nontrivial solutions for nonlinear fractional Schrödinger-Poisson system involving new fractional operator
Keywords:
Fractional Schrödinger-Poisson system, Mountain Pass Theorem, Bessel potential space, perturbation methodAbstract
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.
References
A. Azzollini, P. Alessio, Ground state solutions for the nonlinear Schrödinger-Maxwell equations, J. Math. Anal. Appl., 345 (2008) 90-108.
A. Azzollini, Concentration and compactness in nonlinear Schrödinger-Poisson system with a general nonlinearity, J. Diff. Equa., 249 (2010) 1746-1763.
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.
V. Benci, D. Fortunato, An eigenvalue problem for the Schrödinger-Maxwell equations, Topological Methods Nonlinear Anal., 11 (1998) 283-293.
Z.Binlin, G.M.Bisci, R.Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity, 28 (2015) 2247.
C. Bucur, E. Valdinoci, Nonlocal di?.appl. Cham Springer, 20 (2016).
G. Che, H. Chen, Multiplicity and concentration of solutions for a fractional Schrödinger-Poisson system with sign-changing potential, Appl. Anal., (2021) 1-22.
J. Chen, X. Tang, H. Luo, Infinitely many solutions for fractional Schrödinger-Poisson systems with sign-changing potential, Elec. J. Di?. Equa., 97 (2017).
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.
E. Di Nezza, G. Palatucci, E.Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. scie. math., 136 (2012) 521-573.
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.
X. He, W. Zou, Multiplicity of concentrating positive solutions for Schrödinger-Poisson equations with critical growth, Nonlinear Anal., 170 (2018) 142-170.
R. Jiang, C. Zhai., Two nontrivial solutions for a nonhomogeneous fractional Schrödinger-Poisson equation in R 3 , Boun. Val. Prob., 1 (2020) 1-18.
T. Jin, Multiplicity of solutions for a fractional Schrödinger-Poisson system without (PS) condition, AIMS Math., 6 (2021) 9048-9058.
K. Li, Existence of non-trivial solutions for nonlinear fractional Schrödinger-Poisson equations, Appl. Math. Lett., 72 (2017) 1-9.
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).
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.
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.
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.
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.
T.T. Shieh, D.E. Spector, On a new class of fractional partial differential equations, Advances in Calc. Var., 8 (2015) 321-336.
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.
M. Silhavý, Fractional vector analysis based on invariance requirements (critique of coordinate approaches), Continuum Mech.Thermodynamics, 32 (2020) 207-228.
E. M.Stein, Singular Integrals and Differentiability Properties of Functions (PMS-30), 30. Princeton university press, (2016).
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.
