2000 character limit reached
Infinitely many solutions for a class of fractional Schrodinger equations coupled with neutral scalar field (2402.12006v1)
Published 19 Feb 2024 in math.AP
Abstract: We study the fractional Schr\"{o}dinger equations coupled with a neutral scalar field $$ (-\Delta)s u+V(x)u=K(x)\phi u +g(x)|u|{q-2}u, \quad x\in \mathbb{R}3,\qquad (I-\Delta)t \phi=K(x)u2, \quad x\in \mathbb{R}3, $$ where $(-\Delta)s$ and $(I-\Delta)t$ denote the fractional Laplacian and Bessel operators with $\frac{3}{4} <s<1$ and $0<t<1$, respectively. Under some suitable assumptions for the external potentials $V$, $K$ and $g$, given $q\in(1,2)\cup(2,2_s*)$ with $2_s*:= \frac{6}{3-2s}$, with the help of an improved Fountain theorem dealing with a class of strongly indefinite variational problems approached by Gu-Zhou [Adv. Nonlinear Stud., {\bf 17} (2017), 727--738], we show that the system admits infinitely many nontrivial solutions.
- D.R. Adams, L.I. Hedberg, Function spaces and potential theory, Grundlehren der Mathematischen Wissenschaften 314. Springer-Verlag, Berlin, 1996.
- C.O. Alves, L. Shen, On existence of solutions for some classes of elliptic problems with supercritical exponential growth, Math. Z., (2024), 306:29.
- G. Bisci, V.D. Rădulescu, Ground state solutions of scalar field fractional Schördinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985–3008.
- L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245–1260.
- E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521–573.
- D.E. Edmunds, W.D. Evans, Spectral theory and differential operators, The Clarendon Press, Oxford University Press, New York, 1987.
- Y. Egorov, V. Kondratiev, On Spectral Theory of Elliptic Operators, Birkhäuser, Basel, 1996.
- M.M. Fall, V. Felli, Unique continuation properties for relativistic Schrödinger operators with a singular potential, Discrete Contin. Dyn. Syst., 35 (2015), no. 12, 5827–5867.
- P. Felmer, A. Quaas, J. Tan, Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237–1262.
- P. Felmer, I. Vergara, Scalar field equation with non-local diffusion, Nonlinear Differ. Equ. Appl., 22 (2015), 1411–1428.
- L. Gu, H. Zhou, An improved fountain theorem and its application, Adv. Nonlinear Stud., 17 (2017), 727–738.
- X. He, W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. Partial Differential Equations, 55 (2016), 1–39.
- N. Ikoma, Existence of solutions of scalar field equations with fractional operator, J. Fixed Point Theory Appl., 19 (2017), 649–690.
- W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differential Equations, 3 (1998), 441–472.
- N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298–305.
- N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 56–108.
- E.H. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 1997.
- A. Pankov, Periodic nonlinear Schrödinger equation with application to photonic crystals, Milan J. Math., 73 (2005), 259–287.
- P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270–291.
- S. Secchi, Concave-convex nonlinearities for some nonlinear fractional equations involving the Bessel operator, Complex Var. Elliptic Equ., 62 (2017), 654–669.
- S. Secchi, On some nonlinear fractional equations involving the Bessel operator, J. Dynam. Differential Equations, 29 (2017), 1173–1193.
- S. Secchi, Ground state solutions for nonlinear fractional Schrödinger equations in ℝNsuperscriptℝ𝑁\mathbb{R}^{N}blackboard_R start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT, J. Math. Phys., 54 (2013), 031501.
- L. Shen, Multiplicity and concentration results for fractional Schrödinger-Poisson systems involving a Bessel operator, Math. Methods Appl. Sci., 41 (2018), 7599–7611.
- E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, N.J. 1970.
- A. Szulkin, T. Weth, Ground state solutions for some indefinite variational problems, J. Funct. Anal., 257 (2009), 3802–3822.
- A. Szulkin, T. Weth, The method of Nehari manifold, Handbook of nonconvex analysis and applications, 597–632, Int. Press, Somerville, MA, 2010.
- W.P. Ziemer, Weakly Differentiable Functions, Springer, New York (1989).