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Maximal factors of order $ d $ of dynamical cubespaces
Singular solutions of a Lane-Emden system
1. | Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada |
2. | Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Postal Code: 34149-16818, Qazvin, Iran |
$ \begin{equation} \left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = & \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ \hfill -\Delta u_2 & = & \lambda_2 | \nabla u_1|^q \qquad \mbox{ in } \Omega, \\ \hfill u_1 = u_2 & = & 0 \hfill \mbox{ on } \partial \Omega, \end{array}\right.\;\;\;\;\;\;\;(1) \end{equation} $ |
$ \Omega $ |
$ C^2 $ |
$ B_1 $ |
$ \mathbb{R}^N $ |
$ \lambda_i $ |
$ p $ |
$ q $ |
$ u_1,u_2 $ |
References:
[1] |
A. Aghajani, C. Cowan and S. H. Lui,
Existence and regularity of nonlinear advection problems, Nonlinear Analysis, 166 (2018), 19-47.
doi: 10.1016/j.na.2017.10.007. |
[2] |
A. Aghajani, C. Cowan and S. H. Lui,
Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball, J. Diff. Eqns., 264 (2018), 2865-2896.
doi: 10.1016/j.jde.2017.11.009. |
[3] |
D. Arcoya, L. Boccardo, T. Leonori and A. Porretta,
Some elliptic problems with singular natural growth lower order terms, J. Diff. Eqns., 249 (2010), 2771-2795.
doi: 10.1016/j.jde.2010.05.009. |
[4] |
D. Arcoya, J. Carmona, T. Leonori, P. J. Martinez-Aparicio, L. Orsina and F. Petitta,
Existence and non-existence of solutions for singular quadratic quasilinear equations, J. Diff. Eqns., 246 (2009), 4006-4042.
doi: 10.1016/j.jde.2009.01.016. |
[5] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions, J. Math. Anal. Appl., 420 (2014), 772-780.
doi: 10.1016/j.jmaa.2014.06.007. |
[6] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335.
doi: 10.1016/j.jfa.2015.01.014. |
[7] |
A. Bensoussan, L. Boccardo and F. Murat,
On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 347-364.
doi: 10.1016/S0294-1449(16)30342-0. |
[8] |
M.-F. Bidaut Véron, M. Garcia-Huidobro and L. Véron,
Remarks on some quasilinear equations with gradient terms and measure data, Contemp. Math., 595 (2013), 31-53.
|
[9] |
M.-F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron,
Local and global properties of solutions of quasilinear Hamilton-Jacobi equations, J. Funct. Anal., 267 (2014), 3294-3331.
doi: 10.1016/j.jfa.2014.07.003. |
[10] |
M.-F. Bidaut Véron, M. Garcia-Huidobro and L. Véron,
Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations, Calc. Var., 54 (2015), 3471-3515.
doi: 10.1007/s00526-015-0911-5. |
[11] |
J. Ching and F. Cirstea,
Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term, Anal. PDE, 8 (2015), 1931-1962.
doi: 10.2140/apde.2015.8.1931. |
[12] |
J. M. Coron,
Topologie et cas limite des injections de Sobolev, C.R. Acad. Sc. Paris Sér. I Math., 299 (1984), 209-212.
|
[13] |
C. Cowan and A. Razani, Singular solutions of a $p$-Laplace equation involving the gradient, J. Diff. Eqns., 269 (2020), 3914-3942.
doi: 10.1016/j.jde.2020.03.017. |
[14] |
J. Dávila, M. del Pino and M. Musso,
The supercritical Lane-Emden-Fowler equation in exterior domains, Comm. Partial Differential Equations, 32 (2007), 1225-1243.
doi: 10.1080/03605300600854209. |
[15] |
J. Dávila, M. del Pino, M. Musso and J. Wei,
Fast and slow decay solutions for supercritical elliptic problems in exterior domains, Calc. Var., 32 (2008), 453-480.
doi: 10.1007/s00526-007-0154-1. |
[16] |
J. Dávila, M. del Pino, M. Musso and J. Wei,
Standing waves for supercritical nonlinear Schrödinger equations, J. Diff. Eqns., 236 (2007), 164-198.
doi: 10.1016/j.jde.2007.01.016. |
[17] |
J. Dávila and L. Dupaigne,
Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.
doi: 10.1142/S0219199707002575. |
[18] |
M. del Pino and M. Musso, Super-critical bubbling in elliptic boundary value problems, Variational problems and related topics, (Kyoto, 2002), 1307 (2003), 85-108. |
[19] |
M. del Pino, P. Felmer and M. Musso,
Two-bubble solutions in the super-critical Bahri-Corons problem, Calc. Var., 16 (2003), 113-145.
doi: 10.1007/s005260100142. |
[20] |
M. del Pino, P. Felmer and M. Musso,
Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. London Math. Society, 35 (2003), 513-521.
doi: 10.1112/S0024609303001942. |
[21] |
V. Ferone and F. Murat,
Nonlinear problems having natural growth in the gradient: An existence result when the source terms are small, Nonlinear Anal., 42 (2000), 13309-1326.
doi: 10.1016/S0362-546X(99)00165-0. |
[22] |
V. Ferone, M. R. Posteraro and J. M. Rakotoson, $L^\infty$-estimates for nonlinear elliptic problems with $p$-growth in the gradient, J. Inequal. Appl., 3 (1999), 109-125.. |
[23] |
M. Ghergu and V. D. Rǎdulescu, Nonlinear PDEs, Mathematical Models in Bilology, Chemistry and Popoulation Genetics, Springer-Verlag, Berlin Heidelberg, 2012.
doi: 10.1007/978-3-642-22664-9. |
[24] |
D. Giachetti, F. Petitta and S. Segura de León,
Elliptic equations having a singular quadratic gradient term and a changing sign datum, Commun. Pure Appl. Anal., 11 (2012), 1875-1895.
doi: 10.3934/cpaa.2012.11.1875. |
[25] |
D. Giachetti, F. Petitta and S. Segura de León,
A priori estimates for elliptic problems with a strongly singular gradient term and a general datum, Diff. Integral Eqns., 26 (2013), 913-948.
|
[26] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[27] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, , Clasiics in Mathematics, Springer-Verlag, Berlin, 2001. |
[28] |
N. Grenon, F. Murat and A. Porretta,
Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms, C. R. Acad. Sci. Paris, 342 (2006), 23-28.
doi: 10.1016/j.crma.2005.09.027. |
[29] |
N. Grenon and C. Trombetti,
Existence results for a class of nonlinear elliptic problems with $p$-growth in the gradient, Nonlinear Anal., 52 (2003), 931-942.
doi: 10.1016/S0362-546X(02)00143-8. |
[30] |
J. M. Lasry and P.-L. Lions,
Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Math. Ann., 283 (1989), 583-630.
doi: 10.1007/BF01442856. |
[31] |
D. Lazard,
Quantifier elimination: Optimal solution for two classical examples, J. Symbolic Comput., 5 (1988), 261-266.
doi: 10.1016/S0747-7171(88)80015-4. |
[32] |
P.-L. Lions,
Quelques remarques sur les problemes elliptiques quasilineaires du second ordre, J. Anal. Math., 45 (1985), 234-254.
doi: 10.1007/BF02792551. |
[33] |
M. Marcus and P.-T. Nguyen,
Elliptic equations with nonlinear absorption depending on the solution and its gradient, Proc. London Math. Soc., 111 (2015), 205-239.
doi: 10.1112/plms/pdv020. |
[34] |
R. Mazzeo and F. Pacard,
A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Diff. Geom., 44 (1996), 331-370.
doi: 10.4310/jdg/1214458975. |
[35] |
P.-T. Nguyen,
Isolated singularities of positive solutions of elliptic equations with weighted gradient term, Anal. PDE, 9 (2016), 1671-1692.
doi: 10.2140/apde.2016.9.1671. |
[36] |
P.-T. Nguyen and L. Véron,
Boundary singularities of solutions to elliptic viscous Hamilton-Jacobi equations, J. Funct. Anal., 263 (2012), 1487-1538.
doi: 10.1016/j.jfa.2012.05.019. |
[37] |
F. Pacard and T. Rivière, Linear and Nonlinear Aspects of Vortices: The Ginzburg Landau Model, Progress in Nonlinear Differential Equations and their Applications, 39. Birkhäuser Boston, Inc., Boston, MA, 2000.
doi: 10.1007/978-1-4612-1386-4. |
[38] |
D. Passaseo,
Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., 114 (1993), 97-105.
doi: 10.1006/jfan.1993.1064. |
[39] |
S. Pohozaev,
Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.
|
[40] |
A. Porretta and S. Segura de Leon,
Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl., 85 (2006), 465-492.
doi: 10.1016/j.matpur.2005.10.009. |
[41] |
E. L. Rees, Graphical discussion of the roots of a quartic equation, The American Mathematical Monthly, 29 (1922), 51-55. 10.2307/297280.
doi: 10.1080/00029890.1922.11986100. |
[42] |
M. Struwe, Variational Methods-Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-02624-3. |
[43] |
Z. Zhang,
Boundary blow-up elliptic problems with nonlinear gradient terms, J. Diff. Eqns., 228 (2006), 661-684.
doi: 10.1016/j.jde.2006.02.003. |
show all references
References:
[1] |
A. Aghajani, C. Cowan and S. H. Lui,
Existence and regularity of nonlinear advection problems, Nonlinear Analysis, 166 (2018), 19-47.
doi: 10.1016/j.na.2017.10.007. |
[2] |
A. Aghajani, C. Cowan and S. H. Lui,
Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball, J. Diff. Eqns., 264 (2018), 2865-2896.
doi: 10.1016/j.jde.2017.11.009. |
[3] |
D. Arcoya, L. Boccardo, T. Leonori and A. Porretta,
Some elliptic problems with singular natural growth lower order terms, J. Diff. Eqns., 249 (2010), 2771-2795.
doi: 10.1016/j.jde.2010.05.009. |
[4] |
D. Arcoya, J. Carmona, T. Leonori, P. J. Martinez-Aparicio, L. Orsina and F. Petitta,
Existence and non-existence of solutions for singular quadratic quasilinear equations, J. Diff. Eqns., 246 (2009), 4006-4042.
doi: 10.1016/j.jde.2009.01.016. |
[5] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions, J. Math. Anal. Appl., 420 (2014), 772-780.
doi: 10.1016/j.jmaa.2014.06.007. |
[6] |
D. Arcoya, C. De Coster, L. Jeanjean and K. Tanaka,
Continuum of solutions for an elliptic problem with critical growth in the gradient, J. Funct. Anal., 268 (2015), 2298-2335.
doi: 10.1016/j.jfa.2015.01.014. |
[7] |
A. Bensoussan, L. Boccardo and F. Murat,
On a nonlinear partial differential equation having natural growth terms and unbounded solution, Ann. Inst. H. Poincaré Anal. Non Linéaire, 5 (1988), 347-364.
doi: 10.1016/S0294-1449(16)30342-0. |
[8] |
M.-F. Bidaut Véron, M. Garcia-Huidobro and L. Véron,
Remarks on some quasilinear equations with gradient terms and measure data, Contemp. Math., 595 (2013), 31-53.
|
[9] |
M.-F. Bidaut-Véron, M. Garcia-Huidobro and L. Véron,
Local and global properties of solutions of quasilinear Hamilton-Jacobi equations, J. Funct. Anal., 267 (2014), 3294-3331.
doi: 10.1016/j.jfa.2014.07.003. |
[10] |
M.-F. Bidaut Véron, M. Garcia-Huidobro and L. Véron,
Boundary singularities of positive solutions of quasilinear Hamilton-Jacobi equations, Calc. Var., 54 (2015), 3471-3515.
doi: 10.1007/s00526-015-0911-5. |
[11] |
J. Ching and F. Cirstea,
Existence and classification of singular solutions to nonlinear elliptic equations with a gradient term, Anal. PDE, 8 (2015), 1931-1962.
doi: 10.2140/apde.2015.8.1931. |
[12] |
J. M. Coron,
Topologie et cas limite des injections de Sobolev, C.R. Acad. Sc. Paris Sér. I Math., 299 (1984), 209-212.
|
[13] |
C. Cowan and A. Razani, Singular solutions of a $p$-Laplace equation involving the gradient, J. Diff. Eqns., 269 (2020), 3914-3942.
doi: 10.1016/j.jde.2020.03.017. |
[14] |
J. Dávila, M. del Pino and M. Musso,
The supercritical Lane-Emden-Fowler equation in exterior domains, Comm. Partial Differential Equations, 32 (2007), 1225-1243.
doi: 10.1080/03605300600854209. |
[15] |
J. Dávila, M. del Pino, M. Musso and J. Wei,
Fast and slow decay solutions for supercritical elliptic problems in exterior domains, Calc. Var., 32 (2008), 453-480.
doi: 10.1007/s00526-007-0154-1. |
[16] |
J. Dávila, M. del Pino, M. Musso and J. Wei,
Standing waves for supercritical nonlinear Schrödinger equations, J. Diff. Eqns., 236 (2007), 164-198.
doi: 10.1016/j.jde.2007.01.016. |
[17] |
J. Dávila and L. Dupaigne,
Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.
doi: 10.1142/S0219199707002575. |
[18] |
M. del Pino and M. Musso, Super-critical bubbling in elliptic boundary value problems, Variational problems and related topics, (Kyoto, 2002), 1307 (2003), 85-108. |
[19] |
M. del Pino, P. Felmer and M. Musso,
Two-bubble solutions in the super-critical Bahri-Corons problem, Calc. Var., 16 (2003), 113-145.
doi: 10.1007/s005260100142. |
[20] |
M. del Pino, P. Felmer and M. Musso,
Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. London Math. Society, 35 (2003), 513-521.
doi: 10.1112/S0024609303001942. |
[21] |
V. Ferone and F. Murat,
Nonlinear problems having natural growth in the gradient: An existence result when the source terms are small, Nonlinear Anal., 42 (2000), 13309-1326.
doi: 10.1016/S0362-546X(99)00165-0. |
[22] |
V. Ferone, M. R. Posteraro and J. M. Rakotoson, $L^\infty$-estimates for nonlinear elliptic problems with $p$-growth in the gradient, J. Inequal. Appl., 3 (1999), 109-125.. |
[23] |
M. Ghergu and V. D. Rǎdulescu, Nonlinear PDEs, Mathematical Models in Bilology, Chemistry and Popoulation Genetics, Springer-Verlag, Berlin Heidelberg, 2012.
doi: 10.1007/978-3-642-22664-9. |
[24] |
D. Giachetti, F. Petitta and S. Segura de León,
Elliptic equations having a singular quadratic gradient term and a changing sign datum, Commun. Pure Appl. Anal., 11 (2012), 1875-1895.
doi: 10.3934/cpaa.2012.11.1875. |
[25] |
D. Giachetti, F. Petitta and S. Segura de León,
A priori estimates for elliptic problems with a strongly singular gradient term and a general datum, Diff. Integral Eqns., 26 (2013), 913-948.
|
[26] |
B. Gidas and J. Spruck,
Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.
doi: 10.1002/cpa.3160340406. |
[27] |
D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, , Clasiics in Mathematics, Springer-Verlag, Berlin, 2001. |
[28] |
N. Grenon, F. Murat and A. Porretta,
Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms, C. R. Acad. Sci. Paris, 342 (2006), 23-28.
doi: 10.1016/j.crma.2005.09.027. |
[29] |
N. Grenon and C. Trombetti,
Existence results for a class of nonlinear elliptic problems with $p$-growth in the gradient, Nonlinear Anal., 52 (2003), 931-942.
doi: 10.1016/S0362-546X(02)00143-8. |
[30] |
J. M. Lasry and P.-L. Lions,
Nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints, Math. Ann., 283 (1989), 583-630.
doi: 10.1007/BF01442856. |
[31] |
D. Lazard,
Quantifier elimination: Optimal solution for two classical examples, J. Symbolic Comput., 5 (1988), 261-266.
doi: 10.1016/S0747-7171(88)80015-4. |
[32] |
P.-L. Lions,
Quelques remarques sur les problemes elliptiques quasilineaires du second ordre, J. Anal. Math., 45 (1985), 234-254.
doi: 10.1007/BF02792551. |
[33] |
M. Marcus and P.-T. Nguyen,
Elliptic equations with nonlinear absorption depending on the solution and its gradient, Proc. London Math. Soc., 111 (2015), 205-239.
doi: 10.1112/plms/pdv020. |
[34] |
R. Mazzeo and F. Pacard,
A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Diff. Geom., 44 (1996), 331-370.
doi: 10.4310/jdg/1214458975. |
[35] |
P.-T. Nguyen,
Isolated singularities of positive solutions of elliptic equations with weighted gradient term, Anal. PDE, 9 (2016), 1671-1692.
doi: 10.2140/apde.2016.9.1671. |
[36] |
P.-T. Nguyen and L. Véron,
Boundary singularities of solutions to elliptic viscous Hamilton-Jacobi equations, J. Funct. Anal., 263 (2012), 1487-1538.
doi: 10.1016/j.jfa.2012.05.019. |
[37] |
F. Pacard and T. Rivière, Linear and Nonlinear Aspects of Vortices: The Ginzburg Landau Model, Progress in Nonlinear Differential Equations and their Applications, 39. Birkhäuser Boston, Inc., Boston, MA, 2000.
doi: 10.1007/978-1-4612-1386-4. |
[38] |
D. Passaseo,
Nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains, J. Funct. Anal., 114 (1993), 97-105.
doi: 10.1006/jfan.1993.1064. |
[39] |
S. Pohozaev,
Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.
|
[40] |
A. Porretta and S. Segura de Leon,
Nonlinear elliptic equations having a gradient term with natural growth, J. Math. Pures Appl., 85 (2006), 465-492.
doi: 10.1016/j.matpur.2005.10.009. |
[41] |
E. L. Rees, Graphical discussion of the roots of a quartic equation, The American Mathematical Monthly, 29 (1922), 51-55. 10.2307/297280.
doi: 10.1080/00029890.1922.11986100. |
[42] |
M. Struwe, Variational Methods-Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-662-02624-3. |
[43] |
Z. Zhang,
Boundary blow-up elliptic problems with nonlinear gradient terms, J. Diff. Eqns., 228 (2006), 661-684.
doi: 10.1016/j.jde.2006.02.003. |
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