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Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition
Multiplicity results for fractional systems crossing high eigenvalues
Departamento de Matemática -Instituto de Ciências Exatas, Universidade Federal de Juiz de Fora, 30161-970, Juiz de Fora -MG, Brazil |
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.
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. |
[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. |
[3] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[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. |
[5] |
B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On Some Critical Problems for the fractional Laplacian operator, arXiv: 1106.6081. |
[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. |
[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. |
[8] |
L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems preprint.
doi: 10.3934/nhm.2008.3.523. |
[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. |
[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. |
[11] |
M. Calanchi and B. Ruf,
Elliptic equations with one-sided critical growth, Elect. Journal Diff. Equations, 89 (2002), 1-21.
|
[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. |
[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. |
[14] |
W. Craig, C. Sulem and P. L. Sulem,
Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5 (1992), 497-522.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
F. Gazzola and B. Ruf,
Lower order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Diff. Equations, 4 (1997), 555-572.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
R. Servadei,
The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.
doi: 10.1515/anona-2013-0008. |
[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. |
[31] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Cont. Dyn. Syst., 33 (2013), 2105-2137.
|
[32] |
M. F. Shlesinger, G. M. Zaslavsky and J. Klafter,
Strange kinetics, Nature, 363 (1993), 31-37.
|
[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. |
[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.
|
[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. |
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. |
[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. |
[3] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
|
[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. |
[5] |
B. Barrios, E. Colorado, A. De Pablo and U. Sanchez, On Some Critical Problems for the fractional Laplacian operator, arXiv: 1106.6081. |
[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. |
[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. |
[8] |
L. Caffarelli and A. Mellet, Random homogenization of fractional obstacle problems preprint.
doi: 10.3934/nhm.2008.3.523. |
[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. |
[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. |
[11] |
M. Calanchi and B. Ruf,
Elliptic equations with one-sided critical growth, Elect. Journal Diff. Equations, 89 (2002), 1-21.
|
[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. |
[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. |
[14] |
W. Craig, C. Sulem and P. L. Sulem,
Nonlinear modulation of gravity waves: a rigorous approach, Nonlinearity, 5 (1992), 497-522.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[20] |
F. Gazzola and B. Ruf,
Lower order perturbations of critical growth nonlinearities in semilinear elliptic equations, Adv. Diff. Equations, 4 (1997), 555-572.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[29] |
R. Servadei,
The Yamabe equation in a non-local setting, Adv. Nonlinear Anal., 2 (2013), 235-270.
doi: 10.1515/anona-2013-0008. |
[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. |
[31] |
R. Servadei and E. Valdinoci,
Variational methods for non-local operators of elliptic type, Discrete Cont. Dyn. Syst., 33 (2013), 2105-2137.
|
[32] |
M. F. Shlesinger, G. M. Zaslavsky and J. Klafter,
Strange kinetics, Nature, 363 (1993), 31-37.
|
[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. |
[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.
|
[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. |
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