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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[26]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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