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Quasi-periodic solution of quasi-linear fifth-order KdV equation
On a class of non-local elliptic equations with asymptotically linear term
1. | School of Mathematics, Jilin University, Changchun 130012, China |
2. | School of Mathematical Sciences, Beijing Normal University, No. 19 XinJieKouWai St., HaiDian District, Beijing 100875, China |
We consider the nonlinear elliptic PDE driven by the fractional Laplacian with asymptotically linear term. Some results regarding existence and multiplicity of non-trivial solutions are obtained. More precisely, information about multiple non-trivial solutions is given under some hypotheses of asymptotically linear condition; non-local elliptic equations with combined nonlinearities are also studied, and some results of local existence and global existence are obtained. Finally, an $L^{∞}$ regularity result is also given in the appendix, using the De Giorgi-Stampacchia iteration method.
References:
[1] |
A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[3] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[4] |
X. Cabré and J. -M. Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire, C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366.
doi: 10.1016/j.crma.2009.10.012. |
[5] |
X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[6] |
L. Caffarelli, J. -M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[7] |
L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.
doi: 10.1007/s00526-010-0359-6. |
[8] |
L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[9] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[10] |
D. G. de Figueiredo, J. -P. Gossez and P. Ubilla, Local "superlinearity" and "sublinearity" for the p-Laplacian, J. Funct. Anal., 257 (2009), 721-752.
doi: 10.1016/j.jfa.2009.04.001. |
[11] |
R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344.
doi: 10.1016/j.anihpc.2008.11.002. |
[12] |
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. |
[13] |
S. Dipierro, A. Figalli and E. Valdinoci, Strongly nonlocal dislocation dynamics in crystals, Comm. Partial Differential Equations, 39 (2014), 2351-2387.
doi: 10.1080/03605302.2014.914536. |
[14] |
S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting, Comm. Math. Phys., 333 (2015), 1061-1105.
doi: 10.1007/s00220-014-2118-6. |
[15] |
L. Dupaigne and Y. Sire, A Liouville theorem for non local elliptic equations, in Symmetry for elliptic PDEs, 528 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2010,105-114.
doi: 10.1090/conm/528/10417. |
[16] |
A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods Appl. Sci., 38 (2015), 3551-3563.
doi: 10.1002/mma.3438. |
[17] |
M. d. M. González and R. Monneau,
Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one, Discrete Contin. Dyn. Syst., 32 (2012), 1255-1286.
|
[18] |
Q.-Y. Guan and Z.-M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.
doi: 10.1142/S021949370500150X. |
[19] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[20] |
C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pures Appl. (9), 95 (2011), 72-84.
doi: 10.1016/j.matpur.2010.10.003. |
[21] |
S. Patrizi and E. Valdinoci, Crystal dislocations with different orientations and collisions, Arch. Ration. Mech. Anal., 217 (2015), 231-261.
doi: 10.1007/s00205-014-0832-z. |
[22] |
S. Patrizi and E. Valdinoci, Long-time behavior for crystal dislocation dynamics, Math. Models Methods Appl. Sci., 27 (2017), 2185-2228.
doi: 10.1142/S0218202517500427. |
[23] |
R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[24] |
R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.
doi: 10.3934/cpaa.2013.12.2445. |
[25] |
R. Servadei and E. Valdinoci,
Lewy-stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.
doi: 10.4171/RMI/750. |
[26] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[27] |
R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154, http://projecteuclid.org/euclid.pm/1387570393.
doi: 10.5565/PUBLMAT_58114_06. |
[28] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[29] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
[30] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990, https://doi.org/10.1007/978-3-662-02624-3, Applications to nonlinear partial differential equations and Hamiltonian systems.
doi: 10.1007/978-3-662-03212-1. |
[31] |
J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[32] |
J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Anal., 75 (2012), 2098-2110.
doi: 10.1016/j.na.2011.10.010. |
[33] |
Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124.
doi: 10.1007/s00526-013-0706-5. |
[34] |
M. Willem, Minimax Theorems, 24, Springer, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
Dedicated to Rafael de la Llave on the occasion of his 60th birthday
References:
[1] |
A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[2] |
B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[3] |
C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[4] |
X. Cabré and J. -M. Roquejoffre, Propagation de fronts dans les équations de Fisher-KPP avec diffusion fractionnaire, C. R. Math. Acad. Sci. Paris, 347 (2009), 1361-1366.
doi: 10.1016/j.crma.2009.10.012. |
[5] |
X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093.
doi: 10.1016/j.aim.2010.01.025. |
[6] |
L. Caffarelli, J. -M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math., 63 (2010), 1111-1144.
doi: 10.1002/cpa.20331. |
[7] |
L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differential Equations, 41 (2011), 203-240.
doi: 10.1007/s00526-010-0359-6. |
[8] |
L. A. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461.
doi: 10.1007/s00222-007-0086-6. |
[9] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[10] |
D. G. de Figueiredo, J. -P. Gossez and P. Ubilla, Local "superlinearity" and "sublinearity" for the p-Laplacian, J. Funct. Anal., 257 (2009), 721-752.
doi: 10.1016/j.jfa.2009.04.001. |
[11] |
R. de la Llave and E. Valdinoci, A generalization of Aubry-Mather theory to partial differential equations and pseudo-differential equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1309-1344.
doi: 10.1016/j.anihpc.2008.11.002. |
[12] |
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. |
[13] |
S. Dipierro, A. Figalli and E. Valdinoci, Strongly nonlocal dislocation dynamics in crystals, Comm. Partial Differential Equations, 39 (2014), 2351-2387.
doi: 10.1080/03605302.2014.914536. |
[14] |
S. Dipierro, G. Palatucci and E. Valdinoci, Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting, Comm. Math. Phys., 333 (2015), 1061-1105.
doi: 10.1007/s00220-014-2118-6. |
[15] |
L. Dupaigne and Y. Sire, A Liouville theorem for non local elliptic equations, in Symmetry for elliptic PDEs, 528 of Contemp. Math., Amer. Math. Soc., Providence, RI, 2010,105-114.
doi: 10.1090/conm/528/10417. |
[16] |
A. Fiscella, R. Servadei and E. Valdinoci, Asymptotically linear problems driven by fractional Laplacian operators, Math. Methods Appl. Sci., 38 (2015), 3551-3563.
doi: 10.1002/mma.3438. |
[17] |
M. d. M. González and R. Monneau,
Slow motion of particle systems as a limit of a reaction-diffusion equation with half-Laplacian in dimension one, Discrete Contin. Dyn. Syst., 32 (2012), 1255-1286.
|
[18] |
Q.-Y. Guan and Z.-M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.
doi: 10.1142/S021949370500150X. |
[19] |
N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[20] |
C. Mouhot, E. Russ and Y. Sire, Fractional Poincaré inequalities for general measures, J. Math. Pures Appl. (9), 95 (2011), 72-84.
doi: 10.1016/j.matpur.2010.10.003. |
[21] |
S. Patrizi and E. Valdinoci, Crystal dislocations with different orientations and collisions, Arch. Ration. Mech. Anal., 217 (2015), 231-261.
doi: 10.1007/s00205-014-0832-z. |
[22] |
S. Patrizi and E. Valdinoci, Long-time behavior for crystal dislocation dynamics, Math. Models Methods Appl. Sci., 27 (2017), 2185-2228.
doi: 10.1142/S0218202517500427. |
[23] |
R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[24] |
R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464.
doi: 10.3934/cpaa.2013.12.2445. |
[25] |
R. Servadei and E. Valdinoci,
Lewy-stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.
doi: 10.4171/RMI/750. |
[26] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
|
[27] |
R. Servadei and E. Valdinoci, Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58 (2014), 133-154, http://projecteuclid.org/euclid.pm/1387570393.
doi: 10.5565/PUBLMAT_58114_06. |
[28] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[29] |
Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864.
doi: 10.1016/j.jfa.2009.01.020. |
[30] |
M. Struwe, Variational Methods, Springer-Verlag, Berlin, 1990, https://doi.org/10.1007/978-3-662-02624-3, Applications to nonlinear partial differential equations and Hamiltonian systems.
doi: 10.1007/978-3-662-03212-1. |
[31] |
J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[32] |
J. Tan, Y. Wang and J. Yang, Nonlinear fractional field equations, Nonlinear Anal., 75 (2012), 2098-2110.
doi: 10.1016/j.na.2011.10.010. |
[33] |
Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124.
doi: 10.1007/s00526-013-0706-5. |
[34] |
M. Willem, Minimax Theorems, 24, Springer, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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