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W1, p-estimates for elliptic equations with lower order terms
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 |
$\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. $ |
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. 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. |
[5] |
B. Barrios, E. Colorado, R. 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. Bartolo, V. 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. Bartolo, A. 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. Bolle, N. 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. Candela, G. 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. Dipierro, M. Medina, I. 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. Mellet, S. 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. Pucci, M. 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. Xiang, B.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. Zhang, X. 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. Zhang, X. 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. 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. |
[5] |
B. Barrios, E. Colorado, R. 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. Bartolo, V. 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. Bartolo, A. 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. Bolle, N. 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. Candela, G. 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. Dipierro, M. Medina, I. 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. Mellet, S. 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. Pucci, M. 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. Xiang, B.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. Zhang, X. 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. Zhang, X. 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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