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Global well-posedness for effectively damped wave models with nonlinear memory
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 |
$ \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) $ |
$ a, b\in \mathbb{R} $ |
$ a<b $ |
$ (-\Delta)^{\frac{1}{2}} $ |
$ \mathcal{N}_{1/2} $ |
$ \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]. $ |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing Switzerland, 2016.
doi: 10.1007/978-3-319-28739-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
M. Esteban,
Nonsymmetric ground state of symmetric variational problems, Commun. Pure Appl. Math., XLIV (1991), 259-274.
doi: 10.1002/cpa.3160440205. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[32] |
C. Miranda,
Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital. Ser. II, Anno III, 19 (1940), 5-7.
|
[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. |
[34] |
T. Ozawa,
On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[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. |
[36] |
C. Pozrikidis, The Fractional Laplacian, Taylor & Francis Group, LLC 2016.
doi: 10.1201/b19666. |
[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. |
[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. |
[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. |
[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.
|
[41] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Springer International Publishing Switzerland, 2016.
doi: 10.1007/978-3-319-28739-3. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[23] |
M. Esteban,
Nonsymmetric ground state of symmetric variational problems, Commun. Pure Appl. Math., XLIV (1991), 259-274.
doi: 10.1002/cpa.3160440205. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[32] |
C. Miranda,
Un'osservazione sul teorema di Brouwer, Boll. Unione Mat. Ital. Ser. II, Anno III, 19 (1940), 5-7.
|
[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. |
[34] |
T. Ozawa,
On critical cases of Sobolev's inequalities, J. Funct. Anal., 127 (1995), 259-269.
doi: 10.1006/jfan.1995.1012. |
[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. |
[36] |
C. Pozrikidis, The Fractional Laplacian, Taylor & Francis Group, LLC 2016.
doi: 10.1201/b19666. |
[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. |
[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. |
[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. |
[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.
|
[41] |
M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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