Ground states for fractional nonlocal equations with logarithmic nonlinearity

• Autorzy: Guo Lifeng , Sun Yan , Shi Guannan

Identyfikatory

DOI

10.7494/OpMath.2022.42.2.157

• Języki publikacji: EN

Abstrakty

EN

In this paper, we study on the fractional nonlocal equation with the logarithmic nonlinearity formed by [formula] where 2 < q < 2∗s, LK is a non-local operator, Ω is an open bounded set of Rn with Lipschitz boundary. By using the fractional logarithmic Sobolev inequality and the linking theorem, we present the existence theorem of the ground state solutions for this nonlocal problem.

Słowa kluczowe

EN

linking theorem; ground state; logarithmic nonlinearity; variational methods

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2022
• Tom: Vol. 42, no. 2
• Strony: 157--178
• Opis fizyczny: Bibliogr. 32 poz.

Twórcy

autor

Guo Lifeng

• School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, P.R. China

autor

Sun Yan

• School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, P.R. China

autor

Shi Guannan

• School of Mathematics and Statistics, Northeast Petroleum University, Daqing 163318, P.R. China
• Mathematics and Science College, Shanghai Normal University, Shanghai 200233, P.R. China

Bibliografia

• [1] A.H. Ardila, Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity, Nonlinear Anal. 155 (2019), 52–64.
• [2] L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differetial Equation, Abel Symposia 7 (2012), 37–52.
• [3] L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.
• [4] J. Chao, A. Szulkin, A logarithmic Schrödinger equation with asympototic conditions on the potential, J. Math. Anal. Appl. 437 (2016), 241–254.
• [5] O. Ciaurri, L. Roncal, P.R. Stinga, J.L. Torrea, J.L. Varona, Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications, Adv. Math. 330 (2018), 688–738.
• [6] A. Cotsiolis, N.K. Tavoularis, On logarithmic Sobolev inequalities for higher order fractional derivatives, C. R. Acad. Sci. Paris, Ser. I 340 (2004), 205–208.
• [7] P. D’Avenia, M. Squassina, M. Zenari, Fractional logarithmic Schrödinger equations, Math. Methods Appl. Sci. 38 (2014), 5207–5216.
• [8] Y. Ding, A remark on the linking theorem with applications, Nonlinear Anal. 22 (1994), 237–250.
• [9] R.K. Gettor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc. 101 (1961), 75–90.
• [10] N. Laskin, Fractional quantum mechanics and Levy path integrals, Phys. Lett. A 268 (2000), 298–305.
• [11] S. Liang, P. Pucci, B. Zhang, Multiple solutions for critical Choquard–Kirchhoff type equations, Adv. Nonlinear Anal. 10 (2021), 400–419.
• [12] H. Liu, Z. Liu, Q. Xiao, Ground state solution for a fourth-order nonlinear elliptic problem with logarithmic nonlinearity, Appl. Math. Lett. 79 (2018), 176–181.
• [13] Z. Liu, M. Squassina, J. Zhang, Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension, Nonlinear Differential Equations Appl. 50 (2017), 1–32.
• [14] X. Mingqi, V. Rădulescu, B. Zhang, Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity, Calc. Var. Partial Differential Equations 58 (2019), 1–27.
• [15] J. Mo, Z. Yan, Exitence of solutions to p-Laplace equations with logarithmic nonlinearity, Electron. J. Differential Equations 88 (2009), 1–10.
• [16] G. Molica Bisci, V. Rădulescu, R. Servadei, Variational Methods for Nonlocal Fractional Equations, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2016.
• [17] E.D. Nezza, G. Palatucci, E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
• [18] P. Rabinowitz, Minmax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Rhode Island, USA, 1986.
• [19] R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal. 3 (2013), 235–270.
• [20] R. Servadei, E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 889–898.
• [21] R. Servadei, E. Valdinoci, A Brézis–Nirenberg result for non-local critical equations in low dimension, Commun. Pure. Appl. Anal. 12 (2013), 2445–2464.
• [22] R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Cont. Dyn. A 33 (2013), 2105–2137.
• [23] R. Servadei, E. Valdinoci, The Brézis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2014), 67–102.
• [24] L.X. Truong, The Nehari manifold for fractional p-Laplacian equation with logarithmic nonlinearity on whole space, Comput. Math. with Appl. 78 (2019), 3931–3940.
• [25] M. Xiang, D. Yang, B. Zhang, Degenerate Kirchhoff-type fractional diffusion problem with logarithmic nonlinearity, Asymptotic Anal. 118 (2020), 313–329.
• [26] M. Willem, Minimax Theorems, Birkhäuser Boston, Inc. Boston, MA, 1996.
• [27] T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight functions, J. Math. Anal. Appl. 318 (2006), 253–270.
• [28] M. Xiang, D. Hu, D. Yang, Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity, Nonlinear Anal. 198 (2020), 111899.
• [29] M. Xiang, V. Rădulescu, B. Zhang, Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities, ESAIM: COCV 24 (2018), 1249–1273.
• [30] M. Xiang, B. Zhang, Homoclinic solutions for fractional discrete Laplacian equations, Nonlinear Anal. 198 (2020), 111886.
• [31] M. Xiang, B. Zhang, V. Rădulescu, Superlinear Schrödinger–Kirchhoff type problems involving the fractional p-Laplacian and critical exponent, Adv. Nonlinear Anal. 9 (2020), 690–709.
• [32] P. Zhao, X. Wang, The existence of positive solution of elliptic system by a linking theorem on product space, Nonlinear Anal. 56 (2004), 227–240.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).