2017, 16(3): 823-842. doi: 10.3934/cpaa.2017039

Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators

1. 

School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China

2. 

School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, China

*Corresponding author

Received  May 2016 Revised  January 2017 Published  February 2017

Fund Project: This work is partially supported by the NNSF (No. 11571370,11601508) of China, Natural Science Foundation of Jiangsu Province of China (No. BK20140176), Shandong Provincial Natural Science Foundation, China (No. ZR2014AP011) and the Doctoral Fund of University of Jinan (No.XBS160100118)

In this paper, we consider the following perturbed nonlocal elliptic equation
$\left\{ {\begin{array}{*{20}{l}}{ - {{\cal L}_K}u = \lambda u + f(x,u) + g(x,u),\;\;x \in \Omega ,}\\{u = 0,\;\;x \in \mathbb{R}{^N} \setminus \Omega ,}\end{array}} \right. $
where $\Omega$ is a smooth bounded domain in $\mathbb{R}{^N}$, $\lambda$ is a real parameter and $g$ is a non-odd perturbation term. If $f$ is odd in $u$ and satisfies various superlinear growth conditions at infinity in $u$, infinitely many solutions are obtained in spite of the lack of the symmetry of this problem for any $\lambda\in \mathbb{R}$. The results obtained in this paper may be seen as natural extensions of some classical theorems to the case of nonlocal operators. Moreover, the methods used in this paper can be also applied to obtain some new results for the classical Laplace equation with Dirichlet boundary conditions.
Citation: Liang Zhang, X. H. Tang, Yi Chen. Infinitely many solutions for a class of perturbed elliptic equations with nonlocal operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 823-842. doi: 10.3934/cpaa.2017039
References:
[1]

D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[2]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in RN, J. Differential Equations, 255 (2013), 2340-2362. doi: 10.1016/j.jde.2013.06.016.

[3]

A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32. doi: 10.2307/1998565.

[4]

B. BarriosE. ColoradoA. 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.

[5]

B. BarriosE. ColoradoR. Servadei and F. Soria, A critical fractional equation with concaveconvex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003.

[6]

P. BartoloV. Benci and D. Fortunato, Abstract critical point theorems and application to some nonlinear problems with strong reasonce at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.

[7]

R. BartoloA. M. Candela and A. Salvatore, Infinitely many radial solutions of a nonhomogeneous problem, Discrete Contin. Dyn. Syst. Suppl., (2013), 51-59. doi: 10.3934/proc.2013.2013.51.

[8]

P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288. doi: 10.1006/jdeq.1998.3484.

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P. BolleN. Ghoussoub and H. Tehrani, The multiplicity of solutions in nonhomogeneous boundary boundary value problems, Manuscripta Math., 101 (2002), 325-350. doi: 10.1007/s002290050219.

[10]

Cabré X. 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.

[11]

L. Caffarelli, Nonlocal diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp., 7 (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3.

[12]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[13]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Annals of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[14]

A. M. CandelaG. Palmieri and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27 (2006), 117-132.

[15]

X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027.

[16]

Di Nezza E.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.

[17]

S. DipierroM. MedinaI. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}{^N}$, Manuscripta Math.., (). doi: 10.1007/s00229-016-0878-3.

[18]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.

[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]

X. Q. Liu and F. K. Zhao, Existence of infinitely many solutions for quasilinear elliptic equations perturbed from symmetry, Adv. Nonlinear Studies, 13 (2013), 965-978. doi: 10.1515/ans-2013-0412.

[21]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rational Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.

[22]

R. Metzler and J. Klafter, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161-208. doi: 10.1088/0305-4470/37/31/R01.

[23]

Molica Bisci G. and R. Servadei, A Brezis-Nirenberg splitting approach for nonlocal fractional equations, Nonlinear Anal., 119 (2015), 341-353. doi: 10.1016/j.na.2014.10.025.

[24]

Molica Bisci G. and R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.

[25]

P. PucciM. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous SchrödingerKirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}$N, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.

[26]

P. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769. doi: 10.2307/1998726.

[27]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in CBMS Reg. Conf. Ser. in Math. , vol. 65, Amer. Math. Soc. , Providence, RI, 1986. doi: 10.1090/cbms/065.

[28]

M. Ramos and H. Tehrani, Perturbation from symmetry for indefinite semilinear elliptic equations, Manuscripta Math., 128 (2009), 297-314. doi: 10.1007/s00229-008-0228-1.

[29]

A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Studies, 3 (2003), 1-23. doi: 10.1515/ans-2003-0101.

[30]

M. Schechter and W. Zou, Infinitely many solutions to perturbed elliptic equations, J. Funct. Anal., 228 (2005), 1-38. doi: 10.1016/j.jfa.2005.06.014.

[31]

M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math., 32 (1980), 335-364. doi: 10.1007/BF01299609.

[32]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 1762-1775. doi: 10.1016/j.jmaa.2011.12.032.

[33]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.

[34]

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.

[35]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[36]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Eginburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.

[37]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4.

[38]

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.

[39]

H. T. Tehrani, Infinitely many solutions for indefinite semilinear elliptic equations without symmetry, Comm. Partial Differential Equations, 21 (1996), 541-557. doi: 10.1080/03605309608821196.

[40]

M. Q. XiangB.L. Zhang and V. Rǎdulescu, Existence of solutions for perturbed fractional p-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413. doi: 10.1016/j.jde.2015.09.028.

[41]

S. Yolcu and T. Yolcu, Refined eigenvalue bounds on the Dirichlet fractional Laplacian, J. Math. Phys.,, 56 (2015), 073506. doi: 10.1063/1.4922761.

[42]

L. ZhangX. H. Tang and Y. Chen, Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation, Topol. Methods Nonlinear Anal, 48 (2016), 539-554.

[43]

L. ZhangX. H. Tang and Y. Chen, Infinitely many solutions for indefinite quasilinear Schrödinger equations under broken symmetry situations, Math. Methods Appl. Sci., (). doi: 10.1002/mma.4030.

[44]

L. Zhang and Y. Chen, Infinitely many solutions for sublinear indefinite nonlocal elliptic equations perturbed from symmetry, Nonlinear Anal. TMA, 151 (2017), 126-144. doi: 10.1016/j.na.2016.12.001.

show all references

References:
[1]

D. Applebaum, Lévy processes-from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[2]

G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in RN, J. Differential Equations, 255 (2013), 2340-2362. doi: 10.1016/j.jde.2013.06.016.

[3]

A. Bahri and H. Berestycki, A perturbation method in critical point theory and applications, Trans. Amer. Math. Soc., 267 (1981), 1-32. doi: 10.2307/1998565.

[4]

B. BarriosE. ColoradoA. 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.

[5]

B. BarriosE. ColoradoR. Servadei and F. Soria, A critical fractional equation with concaveconvex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003.

[6]

P. BartoloV. Benci and D. Fortunato, Abstract critical point theorems and application to some nonlinear problems with strong reasonce at infinity, Nonlinear Anal., 7 (1983), 981-1012. doi: 10.1016/0362-546X(83)90115-3.

[7]

R. BartoloA. M. Candela and A. Salvatore, Infinitely many radial solutions of a nonhomogeneous problem, Discrete Contin. Dyn. Syst. Suppl., (2013), 51-59. doi: 10.3934/proc.2013.2013.51.

[8]

P. Bolle, On the Bolza problem, J. Differential Equations, 152 (1999), 274-288. doi: 10.1006/jdeq.1998.3484.

[9]

P. BolleN. Ghoussoub and H. Tehrani, The multiplicity of solutions in nonhomogeneous boundary boundary value problems, Manuscripta Math., 101 (2002), 325-350. doi: 10.1007/s002290050219.

[10]

Cabré X. 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.

[11]

L. Caffarelli, Nonlocal diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symp., 7 (2012), 37-52. doi: 10.1007/978-3-642-25361-4_3.

[12]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[13]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Annals of Math., 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[14]

A. M. CandelaG. Palmieri and A. Salvatore, Radial solutions of semilinear elliptic equations with broken symmetry, Topol. Methods Nonlinear Anal., 27 (2006), 117-132.

[15]

X. J. Chang and Z. Q. Wang, Nodal and multiple solutions of nonlinear problems involving the fractional Laplacian, J. Differential Equations, 256 (2014), 2965-2992. doi: 10.1016/j.jde.2014.01.027.

[16]

Di Nezza E.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.

[17]

S. DipierroM. MedinaI. Peral and E. Valdinoci, Bifurcation results for a fractional elliptic equation with critical exponent in $\mathbb{R}{^N}$, Manuscripta Math.., (). doi: 10.1007/s00229-016-0878-3.

[18]

M. Jara, Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps, Comm. Pure Appl. Math., 62 (2009), 198-214. doi: 10.1002/cpa.20253.

[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]

X. Q. Liu and F. K. Zhao, Existence of infinitely many solutions for quasilinear elliptic equations perturbed from symmetry, Adv. Nonlinear Studies, 13 (2013), 965-978. doi: 10.1515/ans-2013-0412.

[21]

A. MelletS. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Rational Mech. Anal., 199 (2011), 493-525. doi: 10.1007/s00205-010-0354-2.

[22]

R. Metzler and J. Klafter, The restaurant at the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004), 161-208. doi: 10.1088/0305-4470/37/31/R01.

[23]

Molica Bisci G. and R. Servadei, A Brezis-Nirenberg splitting approach for nonlocal fractional equations, Nonlinear Anal., 119 (2015), 341-353. doi: 10.1016/j.na.2014.10.025.

[24]

Molica Bisci G. and R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.

[25]

P. PucciM. Q. Xiang and B. L. Zhang, Multiple solutions for nonhomogeneous SchrödingerKirchhoff type equations involving the fractional p-Laplacian in $\mathbb{R}$N, Calc. Var. Partial Differential Equations, 54 (2015), 2785-2806.

[26]

P. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272 (1982), 753-769. doi: 10.2307/1998726.

[27]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, in CBMS Reg. Conf. Ser. in Math. , vol. 65, Amer. Math. Soc. , Providence, RI, 1986. doi: 10.1090/cbms/065.

[28]

M. Ramos and H. Tehrani, Perturbation from symmetry for indefinite semilinear elliptic equations, Manuscripta Math., 128 (2009), 297-314. doi: 10.1007/s00229-008-0228-1.

[29]

A. Salvatore, Multiple solutions for perturbed elliptic equations in unbounded domains, Adv. Nonlinear Studies, 3 (2003), 1-23. doi: 10.1515/ans-2003-0101.

[30]

M. Schechter and W. Zou, Infinitely many solutions to perturbed elliptic equations, J. Funct. Anal., 228 (2005), 1-38. doi: 10.1016/j.jfa.2005.06.014.

[31]

M. Struwe, Infinitely many critical points for functionals which are not even and applications to superlinear boundary value problems, Manuscripta Math., 32 (1980), 335-364. doi: 10.1007/BF01299609.

[32]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 1762-1775. doi: 10.1016/j.jmaa.2011.12.032.

[33]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.

[34]

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.

[35]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[36]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Eginburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.

[37]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102. doi: 10.1090/S0002-9947-2014-05884-4.

[38]

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.

[39]

H. T. Tehrani, Infinitely many solutions for indefinite semilinear elliptic equations without symmetry, Comm. Partial Differential Equations, 21 (1996), 541-557. doi: 10.1080/03605309608821196.

[40]

M. Q. XiangB.L. Zhang and V. Rǎdulescu, Existence of solutions for perturbed fractional p-Laplacian equations, J. Differential Equations, 260 (2016), 1392-1413. doi: 10.1016/j.jde.2015.09.028.

[41]

S. Yolcu and T. Yolcu, Refined eigenvalue bounds on the Dirichlet fractional Laplacian, J. Math. Phys.,, 56 (2015), 073506. doi: 10.1063/1.4922761.

[42]

L. ZhangX. H. Tang and Y. Chen, Infinitely many solutions for quasilinear Schrödinger equations under broken symmetry situation, Topol. Methods Nonlinear Anal, 48 (2016), 539-554.

[43]

L. ZhangX. H. Tang and Y. Chen, Infinitely many solutions for indefinite quasilinear Schrödinger equations under broken symmetry situations, Math. Methods Appl. Sci., (). doi: 10.1002/mma.4030.

[44]

L. Zhang and Y. Chen, Infinitely many solutions for sublinear indefinite nonlocal elliptic equations perturbed from symmetry, Nonlinear Anal. TMA, 151 (2017), 126-144. doi: 10.1016/j.na.2016.12.001.

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