Infinity weak solutions for a nonlocal Fractional problem with different boundary conditions

Hamza El-houari, Hicham Moussa, Sabiki Hajar

Abstract


In this paper, we investigate two nonlinear equations of fractional type under different external conditions. The first part of this study aims to prove the existence of an infinite number of solutions for nonlocal elliptic problems with non-homogeneous Neumann boundary conditions. The proof is guaranteed by exploiting the correct oscillatory behavior of non-smooth terms. The second section of the paper examines a class of nonlocal elliptic problems in which non-smooth components exhibit a mixed effect of concave and convex nonlinearity at Dirichlet boundary conditions. The nonlinearities do not satisfy Ambrosetti-Rabinowitz and monotonicity conditions. Our framework is a Fractional Orlicz-Sobolev space. To establish the main result, we apply variational approaches paired with Ekeland's variational principle.

Full Text:

PDF

References


G. Abdeljabbar, Multiplicity of Nontrivial Solutions of a Class of Fractional -Laplacian Problem, Zeitschrift fur Analysis und ihre Anwendungen 34 (2015), no. 3, 309{319.

R.A. Adams, J.F. Fournier, Sobolev spaces, Second edition, Elsevier/Academic Press, Amsterdam, 2003.

A. Alberico, A. Cianchi, L. Pick, L. Slavikova, Boundedness of functions in fractional Orlicz-Sobolev spaces, Nonlinear Analysis 230 (2023), 113231.

K.B. Ali, M. Hsini, K. Kefi, N.T. Chung, On a nonlocal fractional p(.,.)-Laplacian problem with competing nonlinearities, Complex Analysis and Operator Theory 13 (2019), no. 3, 1377{1399.

A. Ambrosetti, J.Garcia Azorero, I. Peral, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137 (1996), 219{242.

A. Ambrosetti, H. Brezis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519{543.

E. Arhrrabi, H. El-Houari, Fractional Sobolev space: Study of Kirchhoff-Schrödinger systems with singular nonlinearity, CUBO, A Mathematical Journal 26 (2024), no. 3, 407{430.

E. Azroul, A. Benkirane, M. Srati, M. Shimi, Existence of solutions for a nonlocal Kirchhoff type problem in Fractional Orlicz-Sobolev spaces, arXiv preprint arXiv:1901.05216, (2019).

E. Azroul, A. Benkirane, M. Srati, Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space, Advances in Operator Theory 5 (2020), no. 4, 1599{1617.

E. Azroul, A. Benkirane, M. Srati, Existence of solutions for a non-local type problem in Fractional Orlicz Sobolev Spaces, Adv. Oper. Theory 5 (2020), 1350{1375.

A. Bahrouni, S. Bahrouni, M. Xiang, On a class of nonvariational problems in fractional Orlicz-Sobolev spaces, Nonlinear Analysis 190 (2020), 111595.

S. Bahrouni, A.M. Salort, Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces, ESAIM: Control, Optimisation and Calculus of Variations 27 (2021), S15.

R. Biswas, S. Tiwari, Nehari manifold approach for fractional p(.)-Laplacian system involving concave-convex nonlinearities, Electronic J. of Di_erential Equations 2020 (2020), no. 98, 1{29.

G. Bonanno, G.M. Bisci, V. Radulescu, Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces, Nonlinear Analysis: Theory, Methods & Applications 74 (2011), no. 14, 4785{4795.

G. Bonanno, G.M. Bisci, Infinitely many solutions for a boundary value problem with discontinuous nonlinearities, Boundary Value Problems 2009 (2009), Art. 670675, 1{20.

J.F. Bonder, A.M. Salort, Fractional order Orlicz-Sobolev Spaces, Journal of Functional Analysis 277 (2019), no. 2, 333{367.

J.F. Bonder, A. Salort, H. Vivas, Interior and up to the boundary regularity for the fractional g-Laplacian: the convex case, Nonlinear Analysis 223 (2022), Art. 113060.

A. Boumazourh, M. Srati, Leray-Schauder's solution for a nonlocal problem in a fractional Orlicz-Sobolev space, Moroccan J. of Pure and Appl. Anal. (MJPAA) 6 (2020), no. 1, 42-52.

L.S. Chadli, H. El-Houari, H. Moussa, Multiplicity of solutions for nonlocal parametric elliptic systems in fractional Orlicz-Sobolev spaces, J. of Elliptic and Parabolic Eq. 9 (2023), 1131{1164.

E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions, J. of Functional Analysis 199 (2003), no. 2, 468{507.

D.G. Costa, C.A. Magalhaes, Existence results for perturbations of the p-Laplacian, Nonlinear Analysis: Theory, Methods & Applications 24 (1995), no. 3, 409{418.

E.D. da Silva, M.L.M. Carvalho, J.V. Goncalves, C. Goulart, Critical quasilinear elliptic problems using concave-convex nonlinearities, Annali di Matematica Pura ed Applicata 198 (2019), no. 3, 693{726.

E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des sciences math_ematiques 136 (2012), no. 5, 521{573.

S. Dipierro, X. Ros-Oton, E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Revista Matematica Iberoamericana 33 (2017), no. 2, 377{416.

I. Ekeland, On the variational principle, Journal of Mathematical Analysis and Applications 47 (1974), no. 2, 324-353.

H. El-Houari, L.S. Chadli, H. Moussa, Existence of a solution to a nonlocal Schrödinger system problem in fractional modular spaces, Advances in Operator Theory 7 (2022), no. 1, 1{30.

H. El-Houari, M. Hicham, S. Kassimi, H. Sabiki, Fractional Musielak spaces: a class of nonlocal problem involving concave-convex nonlinearity, J. of Elliptic and Parabolic Eq. 10 (2024), 87{125.

H. El-Houari, L.S. Chadli, M. Hicham, Nehari manifold and fibering map approach for fractional ()-Laplacian Schrödinger system, SeMA Journal 81 (2024), 729{751.

H. El-Houari, L.S. Chadli, H. Moussa, Multiple solutions in fractional Orlicz-Sobolev Spaces for a class of nonlocal Kirchho_ systems, Filomat 38 (2024), no. 8, 2857{2875.

H. El-Houari, H. Moussa, L.S. Chadli, A class of elliptic inclusion in fractional Orlicz-Sobolev spaces, Complex Variables and Elliptic Equations 69 (2022), no. 5, 755{772.

H. El-Houari, L.S. Chadli, H. Moussa, A class of non local elliptic system in non reflexive fractional Orlicz-Sobolev spaces, Asian-European Journal of Mathematics 16 (2023), no. 7, 2350114 (2023).

H. El-Houari, H. Moussa, L.S. Chadli, Ground State Solutions for a Nonlocal System in Fractional Orlicz-Sobolev Spaces, International Journal of Differential Equations 2022 (2022), 1{16.

H. El-Houari, H. Sabiki, H. Moussa, On topological degree for pseudomonotone operators in fractional Orlicz-Sobolev spaces: study of positive solutions of non-local elliptic problems, Advances in Operator Theory 9 (2024), no. 2, Art. 16.

H. El-Houari, L.S. Chadli, H. Moussa, A weak solution to a non-local problem in fractional Orlicz-Sobolev spaces, Asia Pac. J. Math. 10 (2023), no. 2.

H. El-Houari, M. Hicham, H. Sabiki, Multiplicity and concentration properties of solutions for double-phase problem in fractional modular spaces, J. Elliptic Parabol. Equ. 10 (2024), 755{801.

H. El-Houari, H. Moussa, H. Sabiki, Fractional Musielak spaces: a class of non-local elliptic system involving generalized nonlinearity, Rendiconti del Circolo Matematico di Palermo Series 2 (2024), 1{26.

H. El-Houari, H. Moussa, H. Sabiki, Fractional Musielak spaces: study of nonlocal elliptic problem with Choquard-logarithmic nonlinearity, Complex Variables and Elliptic Equations (2024), 1-24.

H. El-Houari, H. Moussa, On a class of generalized Choquard system in fractional Orlicz-Sobolev Spaces, Journal of Mathematical Analysis and Applications 540 (2024), no. 1, 128563.

H. El-Houari, S. Hajar, H. Moussa, Multivalued Elliptic Inclusion in Fractional Orlicz-Sobolev Spaces, Complex Analysis and Operator Theory 18 (2024), no. 4, 94.

N. Fukagai, M. Ito, K. Narukawa, Quasilinear elliptic equations with slowly growing principal part and critical Orlicz-Sobolev nonlinear term, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 139 (2009), no. 1, 73{106.

N. Fukagai, K. Narukawa, On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems, Annali di Matematica Pura ed Applicata 186 (2007), 539{564.

N. Fukagai, M. Ito, K. Narukawa, Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on R, Funkcial. Ekvac. 49 (2006), 235{267.

A. Ghanmi, K. Saoudi, The Nehari manifold for a singular elliptic equation involving the fractional Laplace operator, Fractional Differential Calculus 6 (2016), no. 2, 201{217.

A. Ghanmi, Nontrivial solutions for Kirchhoff-type problems involving the p(x)-Laplace operator, Rocky Mountain J. Math. 48 (2018), no. 4, 1145-1158.

M. Garcia-Huidobro, V.K. Le, R. Manasevich, K. Schmitt, On principal eigenvalues for quasilinear elliptic differential operators: an Orlicz-Sobolev space setting, Nonlinear Differential Equations and Applications NoDEA 6 (1999), no. 2, 207{225.

E.H. Hamza, L.S. Chadli, H. Moussa, Existence of ground state solutions of elliptic system in Fractional Orlicz-Sobolev Spaces, Results in Nonlinear Analysis 5 (2022), no. 2, 112{130.

H. El-houari, L.S. Chadli, H. Moussa, On a class of fractional Kirchhoff-Schrödinger system type, CUBO, A Mathematical Journal 26 (2024), no. 1, 53{73.

S. Heidarkhani, G. Caristi, M. Ferrara, Perturbed Kirchhoff-type Neumann problems in Orlicz-Sobolev spaces, Computers & Mathematics with Applications 71 (2016), no. 10, 2008{2019.

M.A. Krasnosel'skii, Y.B. Rutickii, Convex functions and Orlicz spaces, (Vol. 9), Groningen, Noordhoff, 1961.

A. Kristaly, M. Mihăilescu, V. Rădulescu, Two non-trivial solutions for a non-homogeneous Neumann problem: an Orlicz-Sobolev space setting, Proceedings of the Royal Society of Edinburgh Section A: Mathematics 139 (2009), no. 2, 367{379.

Q.J. Lou, Multiplicity of Positive Solutions for Kirchhoff Systems, Bulletin of the Malaysian Mathematical Sciences Society 43 (2020), no. 5, 3529{3556.

B. Ricceri, A general variational principle and some of its applications, Journal of Computational and Applied Mathematics 113 (2000), no. 1-2, 401{410.

A.M. Salort, Eigenvalues and minimizers for a non-standard growth non-local operator, Journal of Differential Equations 268 (2020), no. 9, 5413{5439.




DOI: https://doi.org/10.52846/ami.v52i1.1902