# American Institute of Mathematical Sciences

February  2021, 41(2): 621-656. doi: 10.3934/dcds.2020291

## 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

* Corresponding author: A. Razani

This work was done when the second author visited the Department of Mathematics, University of Manitoba on a sabbatical leave from Imam Khomeini International University. He appreciates the Department of Mathematics, University of Manitoba, for the hospitality and Prof. C. Cowan for his support.

Received  November 2019 Revised  May 2020 Published  February 2021 Early access  August 2020

In this work we consider the existence of positive singular solutions
 $$$\left\{ \begin{array}{lcl} \hfill -\Delta u_1 & = & \lambda_1 | \nabla u_2|^p \qquad \mbox{ in } \Omega, \\ \hfill -\Delta u_2 & = & \lambda_2 | \nabla u_1|^q \qquad \mbox{ in } \Omega, \\ \hfill u_1 = u_2 & = & 0 \hfill \mbox{ on } \partial \Omega, \end{array}\right.\;\;\;\;\;\;\;(1)$$$
where
 $\Omega$
is small
 $C^2$
perturbation of the unit ball
 $B_1$
in
 $\mathbb{R}^N$
and
 $\lambda_i$
are positive constants. Under suitable conditions on
 $p$
and
 $q$
we prove the existence of positive singular solutions of (1). We also examine the case where one or both of
 $u_1,u_2$
are Hölder continuous.
Citation: Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291
##### 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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