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Regularity criterion of the Newton-Boussinesq equations in $R^3$
Solutions for polyharmonic elliptic problems with critical nonlinearities in symmetric domains
1. | Center for Nonlinear Studies, Northwest University, Xi'an, Shaanxi, 710069, China |
2. | Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi, 710129, China |
References:
[1] |
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. |
[2] |
O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonl. Anal., 133 (1989), 1241-1249.
doi: 10.1016/0362-546X(89)90009-6. |
[3] |
M. Lazzo, Solutions positives multiples pour une équation elliptique non linéaire avec l'exposant critique de Sobolev, C. R. Acad. Sci. Paris Série. I, 314 (1992), 61-64. |
[4] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[5] |
A. Castro and M. Clapp, The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain, Nonlinearity, 16 (2003), 579-590.
doi: 10.1088/0951-7715/16/2/313. |
[6] |
A. Cano and M. Clapp, Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity, J. Diff. Eqs., 237 (2007), 133-158.
doi: 10.1016/j.jde.2007.03.002. |
[7] |
P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83. |
[8] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[9] |
F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Roy. Soc. Edinburgh, 128A (1998), 251-263. |
[10] |
H. Grunau, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order, Analysis, 11 (1991), 83-90.
doi: 10.1080/03605309508821090. |
[11] |
H. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. PDE., 3 (1995), 243-252.
doi: 10.1007/BF01205006. |
[12] |
C. A. Swanson, The best Sobolev constant, Applicable Anal., 47 (1992), 227-239.
doi: 10.1080/00036819208840142. |
[13] |
M. Clapp and M. Squassina, Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data, Comm. Pure Appl.Anal., 2 (2003), 171-186.
doi: 10.3934/cpaa.2003.2.171. |
[14] |
Q. Guo and P. Niu, Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains, J. Diff. Eqs., 245 (2008), 3974-3985.
doi: 10.1016/j.jde.2008.08.002. |
[15] |
M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains, Nonlinear Equations: Methods, Models and Applications, 117-126, Progr. Nonl. Diff. Eqs. Appl., vol. 54, Birkhäuser, Boston, 2003. |
[16] |
D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938.
doi: 10.1090/S0002-9947-04-03769-9. |
[17] |
M. Struwe, "Variational Methods," Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-540-74013-1. |
[18] |
H. Grunau and G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann., 307 (1997), 589-626.
doi: 10.1007/s002080050052. |
[19] |
F. Gazzola, H. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems," Springer, Lecture Notes in Mathematics, 2009, 305-312.
doi: 10.1007/978-3-642-12245-3. |
show all references
References:
[1] |
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. |
[2] |
O. Rey, A multiplicity result for a variational problem with lack of compactness, Nonl. Anal., 133 (1989), 1241-1249.
doi: 10.1016/0362-546X(89)90009-6. |
[3] |
M. Lazzo, Solutions positives multiples pour une équation elliptique non linéaire avec l'exposant critique de Sobolev, C. R. Acad. Sci. Paris Série. I, 314 (1992), 61-64. |
[4] |
G. Cerami, S. Solimini and M. Struwe, Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Anal., 69 (1986), 289-306.
doi: 10.1016/0022-1236(86)90094-7. |
[5] |
A. Castro and M. Clapp, The effect of the domain topology on the number of minimal nodal solutions of an elliptic equation at critical growth in a symmetric domain, Nonlinearity, 16 (2003), 579-590.
doi: 10.1088/0951-7715/16/2/313. |
[6] |
A. Cano and M. Clapp, Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity, J. Diff. Eqs., 237 (2007), 133-158.
doi: 10.1016/j.jde.2007.03.002. |
[7] |
P. Pucci and J. Serrin, Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl., 69 (1990), 55-83. |
[8] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[9] |
F. Gazzola, Critical growth problems for polyharmonic operators, Proc. Roy. Soc. Edinburgh, 128A (1998), 251-263. |
[10] |
H. Grunau, The Dirichlet problem for some semilinear elliptic differential equations of arbitrary order, Analysis, 11 (1991), 83-90.
doi: 10.1080/03605309508821090. |
[11] |
H. Grunau, Positive solutions to semilinear polyharmonic Dirichlet problems involving critical Sobolev exponents, Calc. Var. PDE., 3 (1995), 243-252.
doi: 10.1007/BF01205006. |
[12] |
C. A. Swanson, The best Sobolev constant, Applicable Anal., 47 (1992), 227-239.
doi: 10.1080/00036819208840142. |
[13] |
M. Clapp and M. Squassina, Nonhomogeneous polyharmonic elliptic problems at critical growth with symmetric data, Comm. Pure Appl.Anal., 2 (2003), 171-186.
doi: 10.3934/cpaa.2003.2.171. |
[14] |
Q. Guo and P. Niu, Nodal and positive solutions for singular semilinear elliptic equations with critical exponents in symmetric domains, J. Diff. Eqs., 245 (2008), 3974-3985.
doi: 10.1016/j.jde.2008.08.002. |
[15] |
M. Clapp, A global compactness result for elliptic problems with critical nonlinearity on symmetric domains, Nonlinear Equations: Methods, Models and Applications, 117-126, Progr. Nonl. Diff. Eqs. Appl., vol. 54, Birkhäuser, Boston, 2003. |
[16] |
D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc., 357 (2005), 2909-2938.
doi: 10.1090/S0002-9947-04-03769-9. |
[17] |
M. Struwe, "Variational Methods," Springer-Verlag, Berlin, 1996.
doi: 10.1007/978-3-540-74013-1. |
[18] |
H. Grunau and G. Sweers, Positivity for equations involving polyharmonic operators with Dirichlet boundary conditions, Math. Ann., 307 (1997), 589-626.
doi: 10.1007/s002080050052. |
[19] |
F. Gazzola, H. Grunau and G. Sweers, "Polyharmonic Boundary Value Problems," Springer, Lecture Notes in Mathematics, 2009, 305-312.
doi: 10.1007/978-3-642-12245-3. |
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