# American Institute of Mathematical Sciences

May  2021, 20(5): 2065-2100. doi: 10.3934/cpaa.2021058

## Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition

 1 Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58429-970, Campina Grande - PB - Brazil 2 Departamento de Matemáticas, Universidad Nacional de Trujillo, Av. Juan Pablo II s/n. Trujillo-Peru

* Corresponding author

Received  October 2020 Revised  January 2021 Published  April 2021

Fund Project: C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7 and C. E. Torres Ledesma was partially supported by INC Matemática 88887.136371/2017

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem
 $$$\begin{cases} (-\Delta)^{\frac{1}{2}}u + u = Q(x)f(u)\;\;\mbox{in}\;\;\mathbb{R} \setminus (a, b)\\ \mathcal{N}_{1/2}u(x) = 0\;\;\qquad \qquad \quad \mbox{in}\;\;(a, b), \end{cases}$$ \ \ \ \ \ \ \ \ (0.1)$
where
 $a, b\in \mathbb{R}$
with
 $a , $ (-\Delta)^{\frac{1}{2}} $denotes the fractional Laplacian operator and $ \mathcal{N}_{1/2} $is the nonlocal operator that describes the Neumann boundary condition, which is given by $ \mathcal{N}_{1/2}u(x) = \frac{1}{\pi} \int_{\mathbb{R}\setminus (a, b)} \frac{u(x) - u(y)}{|x-y|^{2}}dy, \;\;x\in [a, b]. $Citation: Claudianor O. Alves, César T. Ledesma. Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2065-2100. doi: 10.3934/cpaa.2021058 ##### References:  [1] C. O. Alves, Existence of a positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in$\mathbb{R}^2$, Milan J. Math., 84 (2016), 1-22. doi: 10.1007/s00032-015-0247-9. Google Scholar [2] C. O. Alves, Multiplicity of solutions for a class of elliptic problem in$\mathbb{R}^2$with Neumann conditions, J. Differ. Equ., 219 (2005), 20-39. doi: 10.1016/j.jde.2004.11.010. Google Scholar [3] C. O. Alves, J. M. do Ó and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in$\mathbb{R}^2$involving critical growth, Nonlinear Anal., 56 (2004), 781-791. doi: 10.1016/j.na.2003.06.003. Google Scholar [4] C. O. Alves, P. C. Carrião and E. S. Medeiros, Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions, Abstr. Appl. Anal.. 3 (2004), 251–268. doi: 10.1155/S1085337504310018. Google Scholar [5] C. O. Alves, G. M. Figueiredo and G. Siciliano, Ground state solutions for fractional scalar field equations under a general critical nonlinearity, Commun. Pure Appl. Anal., 18 (2019), 2199-2215. doi: 10.3934/cpaa.2019099. Google Scholar [6] C. O. Alves, J. M. do Ó and O. H. Miyagaki, Concentration phenomena for fractional elliptic equations involving exponential critical growth, Adv. Nonlinear Stud., 16 (2016), 843-861. doi: 10.1515/ans-2016-0097. Google Scholar [7] C. O. Alves, G. M. Bisci and C. T. Ledesma, Existence of positive solutions for a class of fractional elliptic problem in exterior domain, J. Differ. Equ., 268 (2020), 7183-7219. doi: 10.1016/j.jde.2019.11.068. Google Scholar [8] C. O. Alves and C. T. Ledesma, Fractional elliptic problem in exterior domains with nonlocal Neumann boundary condition, Nonlinear Anal., 195 (2020), 111732. doi: 10.1016/j.na.2019.111732. Google Scholar [9] C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in$\mathbb{R}^N$via penalization method, Calc. Var., 55 (2016), 47. doi: 10.1007/s00526-016-0983-x. Google Scholar [10] C. O. Alves and V. Ambrosio, A multiplicity result for a nonlinear fractional Schrödinger equation in$\mathbb{R}^N$without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 466 (2018), 498-522. doi: 10.1016/j.jmaa.2018.06.005. Google Scholar [11] C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. Google Scholar [12] V. Ambrosio, On a fractional magnetic Schrödinger equation in$\mathbb{R}$with exponential critical growth, Nonlinear Anal., 183 (2019), 117-148. doi: 10.1016/j.na.2019.01.016. Google Scholar [13] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. doi: 10.1007/BF02787822. Google Scholar [14] V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048. Google Scholar [15] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing Switzerland, 2016. doi: 10.1007/978-3-319-28739-3. Google Scholar [16] D. Cao, Multiple solutions for a Neumann problem in an exterior domain, Commun. Partial Differ. Equ., 18 (1993), 687-700. doi: 10.1080/03605309308820945. Google Scholar [17] X. Chang and Z. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differ. Equ., 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027. Google Scholar [18] G. Chen, Singularly perturbed Neumann problem for fractional Schrödinger equations, Sci. China Math., 61 (2018), 695-708. doi: 10.1007/s11425-016-0420-2. Google Scholar [19] F. Demengel and G. Demengel, Functional Spaces for Theory of Elliptic Partial Differential Equations, Springer-Verlag London Limited, 2012. doi: 10.1007/978-1-4471-2807-6. Google Scholar [20] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004. Google Scholar [21] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the whole of$\mathbb{R}^N$, Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8. Google Scholar [22] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416. doi: 10.4171/RMI/942. Google Scholar [23] M. Esteban, Nonsymmetric ground state of symmetric variational problems, Commun. Pure Appl. Math., XLIV (1991), 259-274. doi: 10.1002/cpa.3160440205. Google Scholar [24] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [25] P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var., 54 (2015), 75-98. doi: 10.1007/s00526-014-0778-x. Google Scholar [26] R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in$\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. Google Scholar [27] R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of Radial Solutions for the Fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar [28] A. Iannizzotto and M. Squassina,$1/2$-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385. doi: 10.1016/j.jmaa.2013.12.059. Google Scholar [29] A. Iannizzotto, S. Mosconi and M. Squassina,$H^s$versus$C^0$- weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497. doi: 10.1007/s00030-014-0292-z. Google Scholar [30] H. Kozono, T. Sato and H. Wadade, Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality, Indiana Univ. Math. J., 55 (2006), 1951-1974. doi: 10.1512/iumj.2006.55.2743. Google Scholar [31] S. Lula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical point, Differ. Integral Equ., 29 (2016), 455-492. Google Scholar [32] C. Miranda, Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital. Ser. II, Anno III, 19 (1940), 5-7. Google Scholar [33] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, University Printing House, Cambridge CB2 8BS, United Kingdom, 2016. doi: 10.1017/CBO9781316282397. Google Scholar [34] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012. Google Scholar [35] J. M. do Ó, O. H. Miyagaki and M. Squassina, Non-autonomous fractional problems with exponential growth, Nonlinear Differ. Equ. Appl., 22 (2015), 1395-1410. doi: 10.1007/s00030-015-0327-0. Google Scholar [36] C. Pozrikidis, The Fractional Laplacian, Taylor & Francis Group, LLC 2016. doi: 10.1201/b19666. Google Scholar [37] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. Google Scholar [38] M. Souza and Y. Araújo, On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth, Math. Nachr., 289 (2016), 610-625. doi: 10.1002/mana.201500120. Google Scholar [39] P. Stinga and B. Volzone, Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var., 54 (2015), 1009-1042. doi: 10.1007/s00526-014-0815-9. Google Scholar [40] K. Teng, K. Wang and R. Wang, A sign-changing solution for nonlinear problems involving the fractional Laplacian, Electron. J. Differ. Equ., 2015 (2015), 1-12. Google Scholar [41] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar show all references ##### References:  [1] C. O. Alves, Existence of a positive solution for a nonlinear elliptic equation with saddle-like potential and nonlinearity with exponential critical growth in$\mathbb{R}^2$, Milan J. Math., 84 (2016), 1-22. doi: 10.1007/s00032-015-0247-9. Google Scholar [2] C. O. Alves, Multiplicity of solutions for a class of elliptic problem in$\mathbb{R}^2$with Neumann conditions, J. Differ. Equ., 219 (2005), 20-39. doi: 10.1016/j.jde.2004.11.010. Google Scholar [3] C. O. Alves, J. M. do Ó and O. H. Miyagaki, On nonlinear perturbations of a periodic elliptic problem in$\mathbb{R}^2$involving critical growth, Nonlinear Anal., 56 (2004), 781-791. doi: 10.1016/j.na.2003.06.003. Google Scholar [4] C. O. Alves, P. C. Carrião and E. S. Medeiros, Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions, Abstr. Appl. Anal.. 3 (2004), 251–268. doi: 10.1155/S1085337504310018. Google Scholar [5] C. O. Alves, G. M. Figueiredo and G. Siciliano, Ground state solutions for fractional scalar field equations under a general critical nonlinearity, Commun. Pure Appl. Anal., 18 (2019), 2199-2215. doi: 10.3934/cpaa.2019099. Google Scholar [6] C. O. Alves, J. M. do Ó and O. H. Miyagaki, Concentration phenomena for fractional elliptic equations involving exponential critical growth, Adv. Nonlinear Stud., 16 (2016), 843-861. doi: 10.1515/ans-2016-0097. Google Scholar [7] C. O. Alves, G. M. Bisci and C. T. Ledesma, Existence of positive solutions for a class of fractional elliptic problem in exterior domain, J. Differ. Equ., 268 (2020), 7183-7219. doi: 10.1016/j.jde.2019.11.068. Google Scholar [8] C. O. Alves and C. T. Ledesma, Fractional elliptic problem in exterior domains with nonlocal Neumann boundary condition, Nonlinear Anal., 195 (2020), 111732. doi: 10.1016/j.na.2019.111732. Google Scholar [9] C. O. Alves and O. H. Miyagaki, Existence and concentration of solution for a class of fractional elliptic equation in$\mathbb{R}^N$via penalization method, Calc. Var., 55 (2016), 47. doi: 10.1007/s00526-016-0983-x. Google Scholar [10] C. O. Alves and V. Ambrosio, A multiplicity result for a nonlinear fractional Schrödinger equation in$\mathbb{R}^N$without the Ambrosetti-Rabinowitz condition, J. Math. Anal. Appl., 466 (2018), 498-522. doi: 10.1016/j.jmaa.2018.06.005. Google Scholar [11] C. O. Alves and M. A. S. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153-1166. doi: 10.1007/s00033-013-0376-3. Google Scholar [12] V. Ambrosio, On a fractional magnetic Schrödinger equation in$\mathbb{R}$with exponential critical growth, Nonlinear Anal., 183 (2019), 117-148. doi: 10.1016/j.na.2019.01.016. Google Scholar [13] T. Bartsch, T. Weth and M. Willem, Partial symmetry of least energy nodal solutions to some variational problems, J. Anal. Math., 96 (2005), 1-18. doi: 10.1007/BF02787822. Google Scholar [14] V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 283-300. doi: 10.1007/BF00282048. Google Scholar [15] C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing Switzerland, 2016. doi: 10.1007/978-3-319-28739-3. Google Scholar [16] D. Cao, Multiple solutions for a Neumann problem in an exterior domain, Commun. Partial Differ. Equ., 18 (1993), 687-700. doi: 10.1080/03605309308820945. Google Scholar [17] X. Chang and Z. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differ. Equ., 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027. Google Scholar [18] G. Chen, Singularly perturbed Neumann problem for fractional Schrödinger equations, Sci. China Math., 61 (2018), 695-708. doi: 10.1007/s11425-016-0420-2. Google Scholar [19] F. Demengel and G. Demengel, Functional Spaces for Theory of Elliptic Partial Differential Equations, Springer-Verlag London Limited, 2012. doi: 10.1007/978-1-4471-2807-6. Google Scholar [20] 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. Google Scholar [21] S. Dipierro, M. Medina and E. Valdinoci, Fractional Elliptic Problems with Critical Growth in the whole of$\mathbb{R}^N$, Edizioni della Normale, Pisa, 2017. doi: 10.1007/978-88-7642-601-8. Google Scholar [22] S. Dipierro, X. Ros-Oton and E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 33 (2017), 377-416. doi: 10.4171/RMI/942. Google Scholar [23] M. Esteban, Nonsymmetric ground state of symmetric variational problems, Commun. Pure Appl. Math., XLIV (1991), 259-274. doi: 10.1002/cpa.3160440205. Google Scholar [24] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional laplacian, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 142 (2012), 1237-1262. doi: 10.1017/S0308210511000746. Google Scholar [25] P. Felmer and C. Torres, Non-linear Schrödinger equation with non-local regional diffusion, Calc. Var., 54 (2015), 75-98. doi: 10.1007/s00526-014-0778-x. Google Scholar [26] R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in$\mathbb{R}$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9. Google Scholar [27] R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of Radial Solutions for the Fractional Laplacian, Commun. Pure Appl. Math., 69 (2016), 1671-1726. doi: 10.1002/cpa.21591. Google Scholar [28] A. Iannizzotto and M. Squassina,$1/2$-Laplacian problems with exponential nonlinearity, J. Math. Anal. Appl., 414 (2014), 372-385. doi: 10.1016/j.jmaa.2013.12.059. Google Scholar [29] A. Iannizzotto, S. Mosconi and M. Squassina,$H^s$versus$C^0$- weighted minimizers, Nonlinear Differ. Equ. Appl., 22 (2015), 477-497. doi: 10.1007/s00030-014-0292-z. Google Scholar [30] H. Kozono, T. Sato and H. Wadade, Upper bound of the best constant of a Trudinger-Moser inequality and its application to a Gagliardo-Nirenberg inequality, Indiana Univ. Math. J., 55 (2006), 1951-1974. doi: 10.1512/iumj.2006.55.2743. Google Scholar [31] S. Lula, A. Maalaoui and L. Martinazzi, A fractional Moser-Trudinger type inequality in one dimension and its critical point, Differ. Integral Equ., 29 (2016), 455-492. Google Scholar [32] C. Miranda, Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital. Ser. II, Anno III, 19 (1940), 5-7. Google Scholar [33] G. Molica Bisci, V. Rădulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, University Printing House, Cambridge CB2 8BS, United Kingdom, 2016. doi: 10.1017/CBO9781316282397. Google Scholar [34] T. Ozawa, On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269. doi: 10.1006/jfan.1995.1012. Google Scholar [35] J. M. do Ó, O. H. Miyagaki and M. Squassina, Non-autonomous fractional problems with exponential growth, Nonlinear Differ. Equ. Appl., 22 (2015), 1395-1410. doi: 10.1007/s00030-015-0327-0. Google Scholar [36] C. Pozrikidis, The Fractional Laplacian, Taylor & Francis Group, LLC 2016. doi: 10.1201/b19666. Google Scholar [37] R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. Google Scholar [38] M. Souza and Y. Araújo, On nonlinear perturbations of a periodic fractional Schrödinger equation with critical exponential growth, Math. Nachr., 289 (2016), 610-625. doi: 10.1002/mana.201500120. Google Scholar [39] P. Stinga and B. Volzone, Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var., 54 (2015), 1009-1042. doi: 10.1007/s00526-014-0815-9. Google Scholar [40] K. Teng, K. Wang and R. Wang, A sign-changing solution for nonlinear problems involving the fractional Laplacian, Electron. J. Differ. Equ., 2015 (2015), 1-12. Google Scholar [41] M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-4146-1. Google Scholar  [1] Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105 [2] Maria Francesca Betta, Olivier Guibé, Anna Mercaldo. Uniqueness for Neumann problems for nonlinear elliptic equations. 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