American Institute of Mathematical Sciences

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December  2018, 38(12): 6287-6304. doi: 10.3934/dcds.2018154

On a class of non-local elliptic equations with asymptotically linear term

Yuanhong Wei 1, and  Xifeng Su 2,,

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

* Corresponding author: Xifeng Su

Dedicated to Rafael de la Llave on the occasion of his 60th birthday

Received  September 2017 Revised  November 2017 Published  April 2018

Fund Project: Y. Wei is supported by National Natural Science Foundation of China (Grant No. 11301209) and Outstanding Youth Foundation of Jilin Province of China (Grant No. 20170520056JH), X. Su is supported by National Natural Science Foundation of China (Grant No. 11301513) and "the Fundamental Research Funds for the Central Universities".

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.

Keywords: Non-local operator, variational method, fractional Laplacian, multiplicity.
Mathematics Subject Classification: Primary: 35A15, 35R11; Secondary: 35J67.
Citation: Yuanhong Wei, Xifeng Su. On a class of non-local elliptic equations with asymptotically linear term. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6287-6304. doi: 10.3934/dcds.2018154
References:
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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.

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[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.

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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.

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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.

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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.

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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.

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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.

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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.

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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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