On a Schrödinger–Kirchhoff Type Equation Involving the Fractional p-Laplacian without the Ambrosetti–Rabinowitz Condition
DOI:
https://doi.org/10.15407/mag20.01.041Анотація
У цiй статтi ми розглядаємо iснування та множиннiсть багатьох слабких розв’язкiв для наступного дробового рiвняння типу Шредiнгера–Кiрхгофа: \begin{align*} \left(a+b\displaystyle\iint_{\mathbb{R}^{2N}}\displaystyle\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\mathrm{d}x\mathrm{d}y\right)^{p-1} & \times (-\Delta)^s_pu+\lambda V(x)|u|^{p-2}u \\ & =f(x,u)+h(x)\ \mathrm{ in } \ \mathbb{R}^N,\end{align*} де $N > sp$, $a,b > 0$ — константи, $\lambda$ — параметр, $(-\Delta)^s_p$ — дробовий $p-$оператор Лапласа з $0 < s < 1 < p < \infty$, нелiнiйнiсть $f(x,u)$ i потенцiальна функцiя $V (x)$ задовольняють деякi прийнятнi припущення. За таких умов одержано деякi новi результати для достатньо великих $\lambda >
0$ шляхом застосування варiацiйних методiв.
Mathematical Subject Classification 2020: 35A15, 35J60, 35R11
Ключові слова:
p-оператор Лапласа, дробовий простiр Соболєва, рiвняння типу Шредiнгера–Кiрхгофа, умова Амброзеттi–Рабiновiца, варiацiйнi методиПосилання
R. Lehrer, L.A. Maia, and M. Squassina, On fractional $p$-Laplacian problems with weight, preprint, https://arxiv.org/abs/1404.5452 .
J.M. Mazón, J.D. Rossi, and J. Toledo, Fractional $p$-Laplacian evolution equations, J. Math. Pures Appl. 105 (2016), No. 6, 810--844. https://doi.org/10.1016/j.matpur.2016.02.004
D. Mugnai and E.P. Lippi, Neumann fractional $p$-Laplacian: Eigenvalues and existence results, Nonlinear Anal. 188 (2019), 455--474. https://doi.org/10.1016/j.na.2019.06.015
G. Palatucci, The Dirichlet problem for the $p$-fractional Laplace equation, Nonlinear Anal. 177 (2018), 699--732. https://doi.org/10.1016/j.na.2018.05.004
J. L. Vázquez, The Dirichlet problem for the fractional $p$-Laplacian evolution equation, J. Differential Equations 260 (2016), No. 7, 6038--6056. https://doi.org/10.1016/j.jde.2015.12.033
W. Han and J. Yao, The sign-changing solutions for a class of $p$-Laplacian Kirchhoff type problem in bounded domains, Comput. Math. Appl. 76 (2018), No. 7, 1779--1790. https://doi.org/10.1016/j.camwa.2018.07.029
G. Bonanno and A. Sciammetta, Existence and multiplicity results to Neumann problems for elliptic equations involving the $p$-Laplacian, J. Math. Anal. Appl. 390 (2012), No. 1, 59--67. https://doi.org/10.1016/j.jmaa.2012.01.012
W. Dong and J. Xu, Existence of weak solutions for a $p$-Laplacian problem involving Dirichlet boundary condition, Appl. Math. Comput. 248 (2014), 511--518. https://doi.org/10.1016/j.amc.2014.09.107
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), No. 6, 1842--1864. https://doi.org/10.1016/j.jfa.2009.01.020
M.F. Shlesinger, B.J. West, and J. Klafter, Lévy dynamics of enhanced diffusion: Application to turbulence, Phys. Rev. Lett. 58 (1987), No. 11, 1100--1103. https://doi.org/10.1103/PhysRevLett.58.1100
G.M. Zaslavsky, D. Stevens, and H. Weitzner, Self-similar transport in incomplete chaos, Phys. Rev. E 48 (1993), No. 3, 1683--1694. https://doi.org/10.1103/PhysRevE.48.1683
D. Applebaum, Lévy processes-from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), No. 11, 1336--1347.
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A 268 (2000), No. 4-6, 298--305. https://doi.org/10.1016/S0375-9601(00)00201-2
R. Metzler and J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004), No. 31, R161. https://doi.org/10.1088/0305-4470/37/31/R01
E. D. Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), No. 5, 521--573. https://doi.org/10.1016/j.bulsci.2011.12.004
G. Autuoria and P. Pucci, Elliptic problems involving the fractional Laplacian in $R^{N}$, J. Differential Equations 255 (2013), No. 8, 2340--2362. https://doi.org/10.1016/j.jde.2013.06.016
L. Caffarelli and L. Silvestre, An Extension Problem Related to the Fractional Laplacian, Comm. Partial Differential Equations 32 (2007), No. 8, 1245--1260. https://doi.org/10.1080/03605300600987306
O.H. Miyagaki, S.I. Moreira, and R.S. Vieira, Schrödinger equations involving fractional $p$-Laplacian with supercritical exponent, Complex Var. Elliptic Equ. 67 (2022), No. 5, 1--14. https://doi.org/10.1080/17476933.2020.1857370
J. Zuo, T. An, and M. Li, Superlinear Kirchhoff-type problems of the fractional $p$-Laplacian without the (AR) condition, Bound. Value Probl. 180 (2018), 1--13. https://doi.org/10.1186/s13661-018-1100-1
Q. Chen, C. Chen, and Y. Shi, Multiple solutions for fractional $p$-Laplace equation with concave-convex nonlinearities, Bound. Value Probl. 63 (2020), 1--13. https://doi.org/10.1186/s13661-020-01355-y
M. Xiang, B. Zhang, and V.D. Rădulescu, Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity 29 (2016), No. 10, 3186--3205. https://doi.org/10.1088/0951-7715/29/10/3186
K. Zhang, D. O'Regan, J. Xu, and Z. Fu, Infinitely many solutions via critical points for a fractional $p$-Laplacian equation with perturbations, Adv. Difference Equ. 166 (2019), 1--15. https://doi.org/10.1186/s13662-019-2113-5
Y. Zhang, X. Tang, and J. Zhang, Existence of infinitely many solutions for fractional $p$-Laplacian Schrödinger-Kirchhoff type equations with sign-changing potential, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 113 (2019), No. 2, 569--586. https://doi.org/10.1007/s13398-018-0497-9
P. Pucci, M. Xiang, and B. Zhang, Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional $p$-Laplacian in $R^{N}$, Calc. Var. Partial Differential Equations 54 (2015), No. 3, 2785--2806. https://doi.org/10.1007/s00526-015-0883-5
N. Nyamoradi and L.I. Zaidan, Existence and multiplicity of solutions for fractional $p$-Laplacian Schrödinger-Kirchhoff type equations, Complex Var. Elliptic Equ. 63 (2018), No. 3, 346--359. https://doi.org/10.1080/17476933.2017.1310851
C. Torres, Existence and symmetry result for fractional $p$-Laplacian in $R^{n}$, preprint, https://arxiv.org/abs/1412.3392 .
P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., No. 65, 1986. https://doi.org/10.1090/cbms/065
T. Bartsch, Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), No. 10, 1205--1216. https://doi.org/10.1016/0362-546X(93)90151-H
Z. Binlin, G.M. Bisci, and R. Servadei, Superlinear nonlocal fractional problems with infinitely many solutions, Nonlinearity 28 (2015), 1--18. https://doi.org/10.1088/0951-7715/28/7/2247
T. Bartsch and Z.Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on $R^{N}$, Comm. Partial Differential Equations 20 (1995), No. 9-10, 1725--1741. https://doi.org/10.1080/03605309508821149
Q. Zhang and B. Xu, Multiplicity of solutions for a class of semilinear Schrödinger equations with sign-changing potential, J. Math. Anal. Appl. 377 (2011), No. 2, 834--840. https://doi.org/10.1016/j.jmaa.2010.11.059
X.H. Tang, Infinitely many solutions for semilinear Schrödinger equations with sign-changing potential and nonlinearity, J. Math. Anal. Appl. 401 (2013), No. 1, 407--415. https://doi.org/10.1016/j.jmaa.2012.12.035
