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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

Claudianor O. Alves 1, and  César T. Ledesma 2,,

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{equation} \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} \end{equation} \ \ \ \ \ \ \ \ (0.1) $
where
$ a, b\in \mathbb{R} $
 with
$ a<b $
,
$ (-\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]. $
Keywords: Neumann non local condition, variational methods, nonlinear elliptic equations, nonlocal problems.
Mathematics Subject Classification: Primary: 35A15; Secondary: 35J60, 34B10.
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

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