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The lifespan of solutions to semilinear damped wave equations in one space dimension
The Nehari manifold for fractional systems involving critical nonlinearities
1. | College of Science, Minzu University of China, Beijing 100081, China |
2. | Dipartimento di Informatica, Università degli Studi di Verona, Cá Vignal 2, Strada Le Grazie 15, I-37134 Veron |
3. | Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China |
References:
[1] |
C.O. Alves, D.C. de Morais Filho and M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., 42 (2000), 771-787.
doi: 10.1016/S0362-546X(99)00121-2. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[4] |
B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.
doi: 10.1016/j.anihpc.2014.04.003. |
[5] |
C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A. Math., 142 (2013), 39-71. |
[6] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[7] |
K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[8] |
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. |
[9] |
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. |
[10] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[11] |
W. Chen and S. Deng, The Nehari manifold for a nonlinear elliptic operators involving concave-convex nonlinearities, Z. Angew. Math. Phys., 66, (2015), 1387-1400.
doi: 10.1007/s00033-014-0486-6. |
[12] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[13] |
X. Cheng and S. Ma, Existence of three nontrivial solutions for elliptic systems with critcal exponents and weights, Nonlinear Anal., 69 (2008), 3537-3548.
doi: 10.1016/j.na.2007.09.040. |
[14] |
E. Colorado, A. de Pablo and U. Sánchez, Perturbation of a critical fractional equations, Pacific J. Math., 271 (2014), 65-85.
doi: 10.2140/pjm.2014.271.65. |
[15] |
A. Cotsiolis and N. Tavoularis, Best constant for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[16] |
P. Drabek and S.I. Pohozaev, Positive solutions for the $p$-Laplacian: application of the fibering methods, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.
doi: 10.1017/S0308210500023787. |
[17] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. |
[18] |
L. Faria, O. Miyagaki, F. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103.
doi: 10.1515/anona-2015-0114. |
[19] |
P. Han, The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257. |
[20] |
T. Hsu and H. Lin, Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1163-1177.
doi: 10.1017/S0308210508000875. |
[21] |
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. |
[22] |
X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Comm. Pure Appl. Anal., 13 (2014), 567-584.
doi: 10.3934/cpaa.2014.13.567. |
[23] |
R. Servadei and E. Valdinoci, Mountain pass solutions for nonlinear elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[24] |
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. |
[25] |
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. |
[26] |
J. Serra and X. Ros-Oton, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[27] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[28] |
J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[29] |
Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124.
doi: 10.1007/s00526-013-0706-5. |
[30] |
T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.
doi: 10.1016/j.jmaa.2005.05.057. |
[31] |
T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal., 68 (2008), 1733-1745.
doi: 10.1016/j.na.2007.01.004. |
[32] |
X. Yu, The Nehari manifold for elliptic equation involving the square root of the laplacian, J. Differential Equations, 252 (2012), 1283-1308.
doi: 10.1016/j.jde.2011.09.015. |
show all references
References:
[1] |
C.O. Alves, D.C. de Morais Filho and M.A.S. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal., 42 (2000), 771-787.
doi: 10.1016/S0362-546X(99)00121-2. |
[2] |
A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave-convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.
doi: 10.1006/jfan.1994.1078. |
[3] |
B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[4] |
B. Barrios, E. Colorado, R. Servadei and F. Soria, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900.
doi: 10.1016/j.anihpc.2014.04.003. |
[5] |
C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A. Math., 142 (2013), 39-71. |
[6] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[7] |
K.J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.
doi: 10.1016/S0022-0396(03)00121-9. |
[8] |
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. |
[9] |
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. |
[10] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[11] |
W. Chen and S. Deng, The Nehari manifold for a nonlinear elliptic operators involving concave-convex nonlinearities, Z. Angew. Math. Phys., 66, (2015), 1387-1400.
doi: 10.1007/s00033-014-0486-6. |
[12] |
W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343.
doi: 10.1002/cpa.20116. |
[13] |
X. Cheng and S. Ma, Existence of three nontrivial solutions for elliptic systems with critcal exponents and weights, Nonlinear Anal., 69 (2008), 3537-3548.
doi: 10.1016/j.na.2007.09.040. |
[14] |
E. Colorado, A. de Pablo and U. Sánchez, Perturbation of a critical fractional equations, Pacific J. Math., 271 (2014), 65-85.
doi: 10.2140/pjm.2014.271.65. |
[15] |
A. Cotsiolis and N. Tavoularis, Best constant for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[16] |
P. Drabek and S.I. Pohozaev, Positive solutions for the $p$-Laplacian: application of the fibering methods, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 703-726.
doi: 10.1017/S0308210500023787. |
[17] |
I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. |
[18] |
L. Faria, O. Miyagaki, F. Pereira, M. Squassina and C. Zhang, The Brezis-Nirenberg problem for nonlocal systems, Adv. Nonlinear Anal., 5 (2016), 85-103.
doi: 10.1515/anona-2015-0114. |
[19] |
P. Han, The effect of the domain topology of the number of positive solutions of elliptic systems involving critical Sobolev exponents, Houston J. Math., 32 (2006), 1241-1257. |
[20] |
T. Hsu and H. Lin, Multiple positive solutions for a critical elliptic system with concave-convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1163-1177.
doi: 10.1017/S0308210508000875. |
[21] |
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. |
[22] |
X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Comm. Pure Appl. Anal., 13 (2014), 567-584.
doi: 10.3934/cpaa.2014.13.567. |
[23] |
R. Servadei and E. Valdinoci, Mountain pass solutions for nonlinear elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898.
doi: 10.1016/j.jmaa.2011.12.032. |
[24] |
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. |
[25] |
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. |
[26] |
J. Serra and X. Ros-Oton, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[27] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[28] |
J. Tan, The Brézis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41.
doi: 10.1007/s00526-010-0378-3. |
[29] |
Y. Wei and X. Su, Multiplicity of solutions for non-local elliptic equations driven by the fractional Laplacian, Calc. Var. Partial Differential Equations, 52 (2015), 95-124.
doi: 10.1007/s00526-013-0706-5. |
[30] |
T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.
doi: 10.1016/j.jmaa.2005.05.057. |
[31] |
T.F. Wu, The Nehari manifold for a semilinear elliptic system involving sign-changing weight functions, Nonlinear Anal., 68 (2008), 1733-1745.
doi: 10.1016/j.na.2007.01.004. |
[32] |
X. Yu, The Nehari manifold for elliptic equation involving the square root of the laplacian, J. Differential Equations, 252 (2012), 1283-1308.
doi: 10.1016/j.jde.2011.09.015. |
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