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doi: 10.3934/dcdss.2022130
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## On nonlocal Dirichlet problems with oscillating term

 1 Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, 1000, Slovenia 2 Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Urbino, 61029, Italy 3 Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, 1000, Slovenia

* Corresponding author: Giovanni Molica Bisci

Dedicated to the loving memory of Gaetana Restuccia

Received  April 2021 Revised  May 2022 Early access June 2022

In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401–410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term $f$ has a suitable oscillating behaviour either at the origin or at infinity.

Citation: Boštjan Gabrovšek, Giovanni Molica Bisci, Dušan D. Repovš. On nonlocal Dirichlet problems with oscillating term. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022130
##### References:
 [1] C. Alves and G. Molica Bisci, A compact embedding result for anisotropic Sobolev spaces associated to a strip-like domain and some applications, J. Math. Anal. Appl., 501 (2021), Paper No. 123490, 24 pp. doi: 10.1016/j.jmaa.2019.123490. [2] V. Ambrosio, L. D'Onofrio and and G. Molica Bisci, On nonlocal fractional Laplacian problems with oscillating potentials, Rocky Mountain J. Math., 48 (2018), 1399-1436.  doi: 10.1216/RMJ-2018-48-5-1399. [3] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$–Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737-752.  doi: 10.1017/S0308210509000845. [4] L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian, J. Differential Equations, 263 (2017), 765-778.  doi: 10.1016/j.jde.2017.02.051. [5] F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, Berlin, 2012. doi: 10.1007/978-1-4471-2807-6. [6] G. Devillanova, G. Molica Bisci and R. Servadei, A flower-shape geometry and nonlinear problems on strip-like domains, J. Geom. Anal., 31 (2021), 8105-8143.  doi: 10.1007/s12220-020-00571-3. [7] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 267 (2015), 1807-1836. [8] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [9] F. Faraci, A. Iannizzotto and A. Kristály, Low–dimensional compact embeddings of symmetric Sobolev spaces with applications, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 383-395.  doi: 10.1017/S0308210510000168. [10] G. M. Figueiredo, G. Molica Bisci and R. Servadei, The effect of the domain topology on the number of solutions of fractional Laplace problems, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 103, 24 pp. doi: 10.1007/s00526-018-1382-2. [11] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. [12] A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024. [13] A. Iannizzotto, S. J. N. Mosconi and M. Squassina, Fine boundary regularity for the degenerate fractional $p$-Laplacian, J. Funct. Anal., 279 (2020), 108659, 54 pp. doi: 10.1016/j.jfa.2020.108659. [14] A. Iannizzotto, S. Mosconi and M. Squassina, Sobolev versus Hölder minimizers for the degenerate fractional $p$-Laplacian, Nonlinear Anal., 191 (2020), 111635, 14 pp. doi: 10.1016/j.na.2019.111635. [15] A. Kristály, Infinitely many solutions for a differential inclusion problem in ${ \mathbb R}^N$, J. Differential Equations, 220 (2006), 511-530. [16] A. Kristály, V. D. Rădulescu and C. G. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics. Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631. [17] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1. [18] J. Mawhin and G. Molica Bisci, A Brezis–Nirenberg type result for a non–local fractional operator, J. Lond. Math. Soc., 95 (2017), 73-93.  doi: 10.1112/jlms.12009. [19] G. Molica Bisci, A group–theoretical approach for nonlinear Schrödinger equations, Adv. Calc. Var., 13 (2020), 403-423.  doi: 10.1515/acv-2018-0016. [20] G. Molica Bisci and P. F. Pizzimenti, Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition, Proc. Edinb. Math. Soc., 57 (2014), 779-809.  doi: 10.1017/S0013091513000722. [21] G. Molica Bisci and P. Pucci, Nonlinear Problems with Lack of Compactness, De Gruyter Series in Nonlinear Analysis and Applications, 2021. doi: 10.1515/9783110652017. [22] G. Molica Bisci and V. D. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5. [23] G. Molica Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, No. 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [24] F. Obersnel and P. Omari, Positive solutions of elliptic problems with locally oscillating nonlinearities, J. Math. Anal. Appl., 323 (2006), 913-929.  doi: 10.1016/j.jmaa.2005.11.006. [25] P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Commun. Partial Differential Equations, 21 (1996), 721-733.  doi: 10.1080/03605309608821205. [26] P. Omari and F. Zanolin, An elliptic problem with arbitrarily small positive solutions, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), 301–308, Electron. J. Differ. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000. [27] R. Pei, C. Ma and J. Zhang, Existence results for asymmetric fractional $p$-Laplacian problem, Math. Nachr., 290 (2017), 2673-2683.  doi: 10.1002/mana.201600279. [28] P. Piersanti and P. Pucci, Existence theorems for fractional $p$-Laplacian problems, Anal. Appl. (Singap.), 15 (2017), 607-640.  doi: 10.1142/S0219530516500020. [29] P. Pucci and S. Saldi, Multiple solutions for an eigenvalue problem involving non-local elliptic $p$-Laplacian operators, in Geometric Methods in PDE's, Springer INdAM Series, Vol. 11, G. Citti, M. Manfredini, D. Morbidelli, S. Polidoro, F. Uguzzoni, Eds., Springer, Cham, 2015,159–176. [30] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601. [31] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1.

show all references

##### References:
 [1] C. Alves and G. Molica Bisci, A compact embedding result for anisotropic Sobolev spaces associated to a strip-like domain and some applications, J. Math. Anal. Appl., 501 (2021), Paper No. 123490, 24 pp. doi: 10.1016/j.jmaa.2019.123490. [2] V. Ambrosio, L. D'Onofrio and and G. Molica Bisci, On nonlocal fractional Laplacian problems with oscillating potentials, Rocky Mountain J. Math., 48 (2018), 1399-1436.  doi: 10.1216/RMJ-2018-48-5-1399. [3] G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$–Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737-752.  doi: 10.1017/S0308210509000845. [4] L. M. Del Pezzo and A. Quaas, A Hopf's lemma and a strong minimum principle for the fractional $p$-Laplacian, J. Differential Equations, 263 (2017), 765-778.  doi: 10.1016/j.jde.2017.02.051. [5] F. Demengel and G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, Berlin, 2012. doi: 10.1007/978-1-4471-2807-6. [6] G. Devillanova, G. Molica Bisci and R. Servadei, A flower-shape geometry and nonlinear problems on strip-like domains, J. Geom. Anal., 31 (2021), 8105-8143.  doi: 10.1007/s12220-020-00571-3. [7] A. Di Castro, T. Kuusi and G. Palatucci, Local behavior of fractional $p$-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 267 (2015), 1807-1836. [8] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004. [9] F. Faraci, A. Iannizzotto and A. Kristály, Low–dimensional compact embeddings of symmetric Sobolev spaces with applications, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 383-395.  doi: 10.1017/S0308210510000168. [10] G. M. Figueiredo, G. Molica Bisci and R. Servadei, The effect of the domain topology on the number of solutions of fractional Laplace problems, Calc. Var. Partial Differential Equations, 57 (2018), Paper No. 103, 24 pp. doi: 10.1007/s00526-018-1382-2. [11] G. Franzina and G. Palatucci, Fractional $p$-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. [12] A. Iannizzotto, S. Liu, K. Perera and M. Squassina, Existence results for fractional $p$-Laplacian problems via Morse theory, Adv. Calc. Var., 9 (2016), 101-125.  doi: 10.1515/acv-2014-0024. [13] A. Iannizzotto, S. J. N. Mosconi and M. Squassina, Fine boundary regularity for the degenerate fractional $p$-Laplacian, J. Funct. Anal., 279 (2020), 108659, 54 pp. doi: 10.1016/j.jfa.2020.108659. [14] A. Iannizzotto, S. Mosconi and M. Squassina, Sobolev versus Hölder minimizers for the degenerate fractional $p$-Laplacian, Nonlinear Anal., 191 (2020), 111635, 14 pp. doi: 10.1016/j.na.2019.111635. [15] A. Kristály, Infinitely many solutions for a differential inclusion problem in ${ \mathbb R}^N$, J. Differential Equations, 220 (2006), 511-530. [16] A. Kristály, V. D. Rădulescu and C. G. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics. Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, 136, Cambridge University Press, Cambridge, 2010. doi: 10.1017/CBO9780511760631. [17] E. Lindgren and P. Lindqvist, Fractional eigenvalues, Calc. Var. Partial Differential Equations, 49 (2014), 795-826.  doi: 10.1007/s00526-013-0600-1. [18] J. Mawhin and G. Molica Bisci, A Brezis–Nirenberg type result for a non–local fractional operator, J. Lond. Math. Soc., 95 (2017), 73-93.  doi: 10.1112/jlms.12009. [19] G. Molica Bisci, A group–theoretical approach for nonlinear Schrödinger equations, Adv. Calc. Var., 13 (2020), 403-423.  doi: 10.1515/acv-2018-0016. [20] G. Molica Bisci and P. F. Pizzimenti, Sequences of weak solutions for non-local elliptic problems with Dirichlet boundary condition, Proc. Edinb. Math. Soc., 57 (2014), 779-809.  doi: 10.1017/S0013091513000722. [21] G. Molica Bisci and P. Pucci, Nonlinear Problems with Lack of Compactness, De Gruyter Series in Nonlinear Analysis and Applications, 2021. doi: 10.1515/9783110652017. [22] G. Molica Bisci and V. D. Rădulescu, Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.  doi: 10.1007/s00526-015-0891-5. [23] G. Molica Bisci, V. D. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, No. 162, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397. [24] F. Obersnel and P. Omari, Positive solutions of elliptic problems with locally oscillating nonlinearities, J. Math. Anal. Appl., 323 (2006), 913-929.  doi: 10.1016/j.jmaa.2005.11.006. [25] P. Omari and F. Zanolin, Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential, Commun. Partial Differential Equations, 21 (1996), 721-733.  doi: 10.1080/03605309608821205. [26] P. Omari and F. Zanolin, An elliptic problem with arbitrarily small positive solutions, Proceedings of the Conference on Nonlinear Differential Equations (Coral Gables, FL, 1999), 301–308, Electron. J. Differ. Equ. Conf., 5, Southwest Texas State Univ., San Marcos, TX, 2000. [27] R. Pei, C. Ma and J. Zhang, Existence results for asymmetric fractional $p$-Laplacian problem, Math. Nachr., 290 (2017), 2673-2683.  doi: 10.1002/mana.201600279. [28] P. Piersanti and P. Pucci, Existence theorems for fractional $p$-Laplacian problems, Anal. Appl. (Singap.), 15 (2017), 607-640.  doi: 10.1142/S0219530516500020. [29] P. Pucci and S. Saldi, Multiple solutions for an eigenvalue problem involving non-local elliptic $p$-Laplacian operators, in Geometric Methods in PDE's, Springer INdAM Series, Vol. 11, G. Citti, M. Manfredini, D. Morbidelli, S. Polidoro, F. Uguzzoni, Eds., Springer, Cham, 2015,159–176. [30] V. D. Rădulescu and D. D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015. doi: 10.1201/b18601. [31] B. Ricceri, A general variational principle and some of its applications, J. Comput. Appl. Math., 113 (2000), 401-410.  doi: 10.1016/S0377-0427(99)00269-1.
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