Combined effects for a class of fractional variational inequalities

• Autorzy: Deng Shengbing , Luo Wenshan , Ledesma César E. Torres

Identyfikatory

DOI

10.7494/OpMath.2025.45.2.119

• Języki publikacji: EN

Abstrakty

EN

In this paper, we study the existence of a nonnegative weak solution to the following nonlocal variational inequality: [formula] for all v ∈ K, where s ∈ (0, 1) and M is a continuous steep potential well on RN. Using penalization techniques from del Pino and Felmer, as well as from Bensoussan and Lions, we establish the existence of nonnegative weak solutions. These solutions localize near the potential well Int(M−1(0)).

Słowa kluczowe

EN

fractional variational inequality; variational methods; critical nonlinearity

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2025
• Tom: Vol. 45, no. 2
• Strony: 119--143
• Opis fizyczny: Bibliogr. 42 poz.

Twórcy

autor

Deng Shengbing

• School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China

• https://orcid.org/0000-0003-1136-4398

autor

Luo Wenshan

• School of Mathematics and Statistics, Southwest University, Chongqing 400715, People’s Republic of China

• https://orcid.org/0009-0008-8141-1659

autor

Ledesma César E. Torres

• FCA Research Group, Departamento de Matemáticas, Instituto de Investigación en Matemáticas, Faculta de Ciencias Físicas y Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n, Trujillo, Perú

• https://orcid.org/0000-0002-6889-4982

Bibliografia

• [1] C.O. Alves, L.M. Barros, C. Torres Ledesma, Existence of solution for a class of variational inequality in whole RN with critical growth, J. Math. Anal. Appl. 494 (2021), 124672.
• [2] C.O. Alves, L.M. Barros, C. Torres Ledesma, Existence of solution for a class of variational inequality in whole RN with critical growth: The local mountain pass case, Mediterr. J. Math. 20 (2023), 239.
• [3] C.O. Alves, L.M. Barros, Existence and multiplicity of solutions for a class of elliptic problem with critical growth, Monatsh. Math. 187 (2018), 195–215.
• [4] V. Ambrosio, Multiple concentrating solutions for a fractional Kirchhoff equation with magnetic fields, Discrete Contin. Dyn. Syst. 40 (2020), 781–815.
• [5] V. Ambrosio, Nonlinear Fractional Schrödinger Equations in RN, Frontiers in Elliptic and Parabolic Problems, Birkhäuser Cham, Switzerland AG, 2021.
• [6] T. Bartsch, Z. Wang, Multiple positive solutions for a nonlinear Schrödinger equation, Z. Angew. Math. Phys. 51 (2000), 366–384.
• [7] A. Bensoussan, J.-L. Lions, Applicationa des Inéquationa Variationelles en Contrôle Stochastique, Dunod, Paris, 1978.
• [8] L. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math. 171 (2008), 425–461.
• [9] S. Challal, A. Lyaghfouri, J. Rodrigues, On the A-obstacle problem and the Hausdorff measure of its free boundary, Ann. Mat. Pura Appl. 191 (2012), 113–165.
• [10] Y. Chen, M. Niu, Ground state solutions of nonlinear Schrödinger equations involving the fractional p-Laplacian and potential wells, Open Math. 20 (2022), 50–62.
• [11] M. Clapp, Y. Ding, Positive solutions of a Schrödinger equation with critical nonlinearity, Z. Angew. Math. Phys. 55 (2004), 592–605.
• [12] J.-N. Corvellec, M. Degiovanni, M. Marzocchi, Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal. 1 (1993), 151–171.
• [13] M. Degiovanni, M. Marzocchi, A critical point theory for nonsmooth functionals, Ann. Mat. Pura Appl. 167 (1994), 73–100.
• [14] M. Delpino, P. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. 4 (1996), 121–137.
• [15] S. Deng, W. Luo, C. Torres Ledesma, G. Alama Quiroz, Existence of solutions for a class of fractional Kirchhoff variational inequality, Z. Anal. Anwend. 43 (2024), 149–168.
• [16] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
• [17] Y. Ding, K. Tanaka, Multiplicity of positive solutions of a nonlinear Schrödinger equation, Manuscripta Math. 112 (2003), 109–135.
• [18] M. Fang, Degenerate elliptic inequalities with critical growth, J. Differential Equations 232 (2007), 441–467.
• [19] G.M. Figueiredo, M. Furtado, M. Montenegro, An obstacle problem in a plane domain with two solutions, Adv. Nonlinear Stud. 14 (2014), 327–337.
• [20] S. Frassu, E. Rocha, V. Staicu, The obstacle problem at zero for the fractional p-Laplacian, Set-Valued and Variational Analysis 30 (2022), 207–231.
• [21] L. Guo, T. Hu, Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well, Math. Methods Appl. Sci. 41 (2018), 1145–1161.
• [22] X. He, W. Zou, Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities, Calc. Var. 55 (2016).
• [23] K. Lan, Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities, J. Math. Anal. Appl. 380 (2011), 520–530.
• [24] K. Lan, A variational inequality theory in reflexive smooth Banach spaces and applications to p-Lapacian elliptic inequalities, Nonlinear Anal. 113 (2015), 71–86.
• [25] G. Li, Some properties of weak solutions of nonlinear scalar field equations, Ann. Acad. Sci. Fenn. Ser. A I Math. 15 (1990), 27–36.
• [26] C. Lo, J. Rodrigues, On a class of nonlocal obstacle type problems related to the distributional Riesz fractional derivative, Port. Math. 80 (2023), 157–205.
• [27] A. Loffe, E. Schwartzman, Metric critical point theory I: Morse regularity and homotopic stability of a minimum, J. Math. Pures Appl. 75 (1996), 125–153.
• [28] P. Ma, J. Zhang, Existence and multiplicity of solutions for fractional Choquard equations, Nonlinear Anal. 164 (2017), 100–117.
• [29] P. Magrone, D. Mugnai, R. Servadei, Multiplicity of solutions for semilinear variational inequalities via linking and ∇-theorems, J. Differential Equations 228 (2006), 191–225.
• [30] M. Matzeu, R. Servadei, Semilinear elliptic variational inequalities with dependence on the gradient via mountain pass techniques, Nonlinear Anal. 72 (2010), 4347–4359.
• [31] M. Matzeu, R. Servadei, Stability for semilinear elliptic variational inequalities depending on the gradient, Nonlinear Anal. 74 (2011), 5161–5170.
• [32] P. Panagiotopoulos, Hemivariational Inequalities: Applications to Mechanics and Engineering, Springer, New York, 1993.
• [33] J. Rodrigues, Obstacle Problems in Mathematical Physics, vol. 134, North-Holland Mathematics Studies, Notas de Matemática (Mathematical Notes), Amsterdam, 1987.
• [34] R. Servadei, E. Valdinoci, Lewy–Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam. 29 (2013), 1091–1126.
• [35] L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112.
• [36] K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Anal. Real World Appl. 14 (2013), 867–874.
• [37] C. Torres Ledesma, Existence and concentration of solutions for a non-linear fractional Schrödinger equation with steep potential well, Comm. Pure and Appl. Anal. 15 (2016), 535–547.
• [38] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
• [39] M. Xiang, B. Zhang, H. Qiu, Existence of solutions for a critical fractional Kirchhoff type problem in RN, Sci. China Math. 60 (2017), 1647–1660.
• [40] J. Yang, Positive solutions of an obstacle problem, Ann. Fac. Sci. Toulouse Math. 4 (1995), 339–366.
• [41] J. Yang, Positive solutions of quasilinear elliptic obstacle problems with critical exponents, Nonlinear Anal. 25 (1995), 1283–1306.
• [42] L. Yang, Z. Liu, Multiplicity and concentration of solutions for fractional Schrödinger equation with sublinear perturbation and steep potential well, Comput. Math. Appl. 72 (2013), 1629–1640.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025)