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November  2017, 16(6): 2069-2088. doi: 10.3934/cpaa.2017102

Multiplicity results for fractional systems crossing high eigenvalues

Fábio R. Pereira

Departamento de Matemática -Instituto de Ciências Exatas, Universidade Federal de Juiz de Fora, 30161-970, Juiz de Fora -MG, Brazil

Received  December 2016 Revised  May 2017 Published  July 2017

Fund Project: F. R. Pereira was supported by the Program CAPES/Brazil (Proc 99999.007090/2014-05) at the Granada University and partially by Fapemig/Brazil (CEX APQ 00972/13).

We investigate the existence of solutions for a system of nonlocal equations involving the fractional Laplacian operator and with nonlinearities reaching the subcritical growth and interacting, in some sense, with the spectrum of the operator.

Keywords: Fractional systems, Mountain Pass Theorem, Linking Theorem, resonance, variational method.
Mathematics Subject Classification: Primary: 35R11; Secondary: 35B33, 35J50, 35B34.
Citation: Fábio R. Pereira. Multiplicity results for fractional systems crossing high eigenvalues. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2069-2088. doi: 10.3934/cpaa.2017102
References:
[1]

C. O. Alves, D. C. de Morais Filho and O. H. Miyagaki, Multiple solutions for an elliptic system on bounded and unbounded domains, Nonlinear Anal., 56 (2004), 555-568.  doi: 10.1016/j.na.2003.10.004.  Google Scholar

[2]

A. Ambrosetti and G. Prodi, On the inversion of some differential mappings with singularities between Banach spaces, Ann. Mat. Pura. Appl., 93 (1972), 231-246.  doi: 10.1007/BF02412022.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.   Google Scholar

[4]

D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at $-∞$ and superlinear at $+ ∞$, Math. Z., 219 (1995), 499-513.  doi: 10.1007/BF02572378.  Google Scholar

[5]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On Some Critical Problems for the fractional Laplacian operator, arXiv: 1106.6081. Google Scholar

[6]

M. Bouchekif and Y. Nasri, On a singular elliptic system at resonance, Annali Mat., 189 (2010), 227-240.  doi: 10.1007/s10231-009-0106-9.  Google Scholar

[7]

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.  Google Scholar

[8]

L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems preprint. doi: 10.3934/nhm.2008.3.523.  Google Scholar

[9]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian Inventiones Math. (2007), online. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[10]

L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

[11]

M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Elect. Journal Diff. Equations, 89 (2002), 1-21.   Google Scholar

[12]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.  Google Scholar

[13]

D. G. Costa and A. C. Maganhaes, A variational approach to subquadratic perturbations of elliptic systems, J. Diff. Equations, 111 (1994), 103-122.  doi: 10.1006/jdeq.1994.1077.  Google Scholar

[14]

W. Craig, C. Sulem and P. L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5 (1992), 497-522.   Google Scholar

[15]

L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang, The Brézis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2015), 85-103.  doi: 10.1515/anona-2015-0114.  Google Scholar

[16]

D. G. de Figueiredo and J. Yang, Critical superlinear Ambrosetti-Prodi problems, Top. Methods Nonlinear Anal., 14 (1999), 59-80.  doi: 10.12775/TMNA.1999.022.  Google Scholar

[17]

D. C. de Morais Filho and F. R. Pereira, Critical Ambrosetti-Prodi type problems for systems of elliptic equations, Nonlinear Analysis Theory, Meth. and App., 68 (2008), 194-207.  doi: 10.1016/j.na.2006.10.041.  Google Scholar

[18]

D. C. de Morais Filho and M. A. S. Souto, Systems of p-laplacean equations involving homogeneous nonlinearities with critical sobolev exponent degrees, Commun. in Partial Diff. Equations, 24 (1999), 1537-1553.  doi: 10.1080/03605309908821473.  Google Scholar

[19]

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.  Google Scholar

[20]

F. Gazzola and B. Ruf, Lower order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Diff. Equations, 4 (1997), 555-572.   Google Scholar

[21]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional laplacian to appear in Math. Res. Lett. doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[22]

K. Ito, Lectures on stochastic processes, Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Notes by K. Muralidhara Rao. 2nd edition, Bombay, 24 1984.  Google Scholar

[23]

F. R. Pereira, Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents, Comm. Pure Appl. Analysis., 7 (2008), 355-372.  doi: 10.3934/cpaa.2008.7.355.  Google Scholar

[24]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS, American Mathematical Society, 65 1986. doi: 10.1090/cbms/065.  Google Scholar

[25]

B. Ribeiro, The Ambrosetti-Prodi problem for gradient elliptic systems with critical homogeneous nonlinearity, J. Math. Anal. Appl., 363 (2010), 606-617.  doi: 10.1016/j.jmaa.2009.09.048.  Google Scholar

[26]

X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var., 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1.  Google Scholar

[27]

B. Ruf and P. N. Srikanth, Multiplicity results for superlinear elliptic problems with partial interference with the spectrum, J. Math. Anal. App., 118 (1986), 15-23.  doi: 10.1016/0022-247X(86)90286-6.  Google Scholar

[28]

O. Savin and E. Valdinoci, $Γ$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.  doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[29]

R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.  doi: 10.1515/anona-2013-0008.  Google Scholar

[30]

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

[31]

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

[32]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37.   Google Scholar

[33]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[34]

T. H. Solomon, E. R. Weeks and H. L. Swinney, Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow, Physic. Rew. Lett., 71 (1993), 3975-3978.   Google Scholar

[35]

G. M. Viswanathan, E. P. Raposo and M. G. E. Da Luz, Lévy flights and superdiffusion in the context of biological encounters and random searches, Phys. Life Rev., 5 (2008), 133-150.  doi: 10.1088/1751-8113/42/43/434003.  Google Scholar

show all references

References:
[1]

C. O. Alves, D. C. de Morais Filho and O. H. Miyagaki, Multiple solutions for an elliptic system on bounded and unbounded domains, Nonlinear Anal., 56 (2004), 555-568.  doi: 10.1016/j.na.2003.10.004.  Google Scholar

[2]

A. Ambrosetti and G. Prodi, On the inversion of some differential mappings with singularities between Banach spaces, Ann. Mat. Pura. Appl., 93 (1972), 231-246.  doi: 10.1007/BF02412022.  Google Scholar

[3]

A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.   Google Scholar

[4]

D. Arcoya and S. Villegas, Nontrivial solutions for a Neumann problem with a nonlinear term asymptotically linear at $-∞$ and superlinear at $+ ∞$, Math. Z., 219 (1995), 499-513.  doi: 10.1007/BF02572378.  Google Scholar

[5]

B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On Some Critical Problems for the fractional Laplacian operator, arXiv: 1106.6081. Google Scholar

[6]

M. Bouchekif and Y. Nasri, On a singular elliptic system at resonance, Annali Mat., 189 (2010), 227-240.  doi: 10.1007/s10231-009-0106-9.  Google Scholar

[7]

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.  Google Scholar

[8]

L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems preprint. doi: 10.3934/nhm.2008.3.523.  Google Scholar

[9]

L. Caffarelli, S. Salsa and L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian Inventiones Math. (2007), online. doi: 10.1007/s00222-007-0086-6.  Google Scholar

[10]

L. A. Caffarelli and P. E. Souganidis, Convergence of nonlocal threshold dynamics approximations to front propagation, Arch. Ration. Mech. Anal., 195 (2010), 1-23.  doi: 10.1007/s00205-008-0181-x.  Google Scholar

[11]

M. Calanchi and B. Ruf, Elliptic equations with one-sided critical growth, Elect. Journal Diff. Equations, 89 (2002), 1-21.   Google Scholar

[12]

A. Córdoba and D. Córdoba, A maximum principle applied to quasi-geostrophic equations, Comm. Math. Phys., 249 (2004), 511-528.  doi: 10.1007/s00220-004-1055-1.  Google Scholar

[13]

D. G. Costa and A. C. Maganhaes, A variational approach to subquadratic perturbations of elliptic systems, J. Diff. Equations, 111 (1994), 103-122.  doi: 10.1006/jdeq.1994.1077.  Google Scholar

[14]

W. Craig, C. Sulem and P. L. Sulem, Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5 (1992), 497-522.   Google Scholar

[15]

L. F. O. Faria, O. H. Miyagaki, F. R. Pereira, M. Squassina and C. Zhang, The Brézis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2015), 85-103.  doi: 10.1515/anona-2015-0114.  Google Scholar

[16]

D. G. de Figueiredo and J. Yang, Critical superlinear Ambrosetti-Prodi problems, Top. Methods Nonlinear Anal., 14 (1999), 59-80.  doi: 10.12775/TMNA.1999.022.  Google Scholar

[17]

D. C. de Morais Filho and F. R. Pereira, Critical Ambrosetti-Prodi type problems for systems of elliptic equations, Nonlinear Analysis Theory, Meth. and App., 68 (2008), 194-207.  doi: 10.1016/j.na.2006.10.041.  Google Scholar

[18]

D. C. de Morais Filho and M. A. S. Souto, Systems of p-laplacean equations involving homogeneous nonlinearities with critical sobolev exponent degrees, Commun. in Partial Diff. Equations, 24 (1999), 1537-1553.  doi: 10.1080/03605309908821473.  Google Scholar

[19]

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.  Google Scholar

[20]

F. Gazzola and B. Ruf, Lower order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Diff. Equations, 4 (1997), 555-572.   Google Scholar

[21]

A. Greco and R. Servadei, Hopf's lemma and constrained radial symmetry for the fractional laplacian to appear in Math. Res. Lett. doi: 10.4310/MRL.2016.v23.n3.a14.  Google Scholar

[22]

K. Ito, Lectures on stochastic processes, Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Notes by K. Muralidhara Rao. 2nd edition, Bombay, 24 1984.  Google Scholar

[23]

F. R. Pereira, Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents, Comm. Pure Appl. Analysis., 7 (2008), 355-372.  doi: 10.3934/cpaa.2008.7.355.  Google Scholar

[24]

P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS, American Mathematical Society, 65 1986. doi: 10.1090/cbms/065.  Google Scholar

[25]

B. Ribeiro, The Ambrosetti-Prodi problem for gradient elliptic systems with critical homogeneous nonlinearity, J. Math. Anal. Appl., 363 (2010), 606-617.  doi: 10.1016/j.jmaa.2009.09.048.  Google Scholar

[26]

X. Ros-Oton and J. Serra, The extremal solution for the fractional Laplacian, Calc. Var., 50 (2014), 723-750.  doi: 10.1007/s00526-013-0653-1.  Google Scholar

[27]

B. Ruf and P. N. Srikanth, Multiplicity results for superlinear elliptic problems with partial interference with the spectrum, J. Math. Anal. App., 118 (1986), 15-23.  doi: 10.1016/0022-247X(86)90286-6.  Google Scholar

[28]

O. Savin and E. Valdinoci, $Γ$-convergence for nonlocal phase transitions, Ann. Inst. H. Poincaré Anal. Non Linéaire, 29 (2012), 479-500.  doi: 10.1016/j.anihpc.2012.01.006.  Google Scholar

[29]

R. Servadei, The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.  doi: 10.1515/anona-2013-0008.  Google Scholar

[30]

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

[31]

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

[32]

M. F. Shlesinger, G. M. Zaslavsky and J. Klafter, Strange kinetics, Nature, 363 (1993), 31-37.   Google Scholar

[33]

L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2006), 67-112.  doi: 10.1002/cpa.20153.  Google Scholar

[34]

T. H. Solomon, E. R. Weeks and H. L. Swinney, Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow, Physic. Rew. Lett., 71 (1993), 3975-3978.   Google Scholar

[35]

G. M. Viswanathan, E. P. Raposo and M. G. E. Da Luz, Lévy flights and superdiffusion in the context of biological encounters and random searches, Phys. Life Rev., 5 (2008), 133-150.  doi: 10.1088/1751-8113/42/43/434003.  Google Scholar

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