January  2022, 21(1): 141-158. doi: 10.3934/cpaa.2021172

Singular solutions of a Hénon equation involving a nonlinear gradient term

1. 

Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada

2. 

Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, 34149-16818, Qazvin, Iran

* Corresponding author

Received  March 2021 Revised  September 2021 Published  January 2022 Early access  September 2021

Here, we consider positive singular solutions of
$ \begin{equation*} \left\{ \begin{array}{lcc} -\Delta u = |x|^\alpha |\nabla u|^p & \text{in}& \Omega \backslash\{0\},\\ u = 0&\text{on}& \partial \Omega, \end{array} \right. \end{equation*} $
where
$ \Omega $
is a small smooth perturbation of the unit ball in
$ \mathbb{R}^N $
and
$ \alpha $
and
$ p $
are parameters in a certain range. Using an explicit solution on
$ B_1 $
and a linearization argument, we obtain positive singular solutions on perturbations of the unit ball.
Citation: Craig Cowan, Abdolrahman Razani. Singular solutions of a Hénon equation involving a nonlinear gradient term. Communications on Pure & Applied Analysis, 2022, 21 (1) : 141-158. doi: 10.3934/cpaa.2021172
References:
[1]

A. AghajaniC. Cowan and S. H. Lui, Existence and regularity of nonlinear advection problems, Nonlinear Anal., 166 (2018), 19-47.  doi: 10.1016/j.na.2017.10.007.  Google Scholar

[2]

A. AghajaniC. Cowan and S. H. Lui, Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball, J. Differ. Equ., 264 (2018), 2865-2896.  doi: 10.1016/j.jde.2017.11.009.  Google Scholar

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J. Ching and F. C. 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.  Google Scholar

[4]

C. Cowan and A. Razani, Singular solutions of a $p$-Laplace equation involving the gradient, J. Differ. Equ., 269 (2020), 3914-3942.  doi: 10.1016/j.jde.2020.03.017.  Google Scholar

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C. Cowan and A. Razani, Singular solutions of a Lane-Emden system, Discrete Contin. Dyn. Syst., 41 (2021), 621-656.  doi: 10.3934/dcds.2020291.  Google Scholar

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J. DávilaM. del Pino and M. Musso, The Supercritical Lane-Emden-Fowler Equation in Exterior Domains, Commun. Partial Differ. Equ., 32 (2007), 1225-1243.  doi: 10.1080/03605300600854209.  Google Scholar

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J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.  doi: 10.1142/S0219199707002575.  Google Scholar

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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.  Google Scholar

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M. del PinoP. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron problem, Calc. Var. Partial Differ. Equ., 16 (2003), 113-145.  doi: 10.1007/s005260100142.  Google Scholar

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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.  Google Scholar

[11]

M. Gherga and V. Rădulescu, Nonlinear PDEs, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-22664-9.  Google Scholar

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B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[13]

N. GrenonF. Murat and A. Porretta, Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 23-28.  doi: 10.1016/j.crma.2005.09.027.  Google Scholar

[14]

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.  Google Scholar

[15]

R. Mazzeo and F. Pacard, A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Differ. Geom., 44 (1996), 331-370.   Google Scholar

[16]

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.  Google Scholar

[17]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.   Google Scholar

[18]

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.  Google Scholar

[19]

M. Struwe, Variational Methods–Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Berlin, Springer-Verlag, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[20]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms, J. Differ. Equ., 228 (2006), 661-684.  doi: 10.1016/j.jde.2006.02.003.  Google Scholar

show all references

References:
[1]

A. AghajaniC. Cowan and S. H. Lui, Existence and regularity of nonlinear advection problems, Nonlinear Anal., 166 (2018), 19-47.  doi: 10.1016/j.na.2017.10.007.  Google Scholar

[2]

A. AghajaniC. Cowan and S. H. Lui, Singular solutions of elliptic equations involving nonlinear gradient terms on perturbations of the ball, J. Differ. Equ., 264 (2018), 2865-2896.  doi: 10.1016/j.jde.2017.11.009.  Google Scholar

[3]

J. Ching and F. C. 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.  Google Scholar

[4]

C. Cowan and A. Razani, Singular solutions of a $p$-Laplace equation involving the gradient, J. Differ. Equ., 269 (2020), 3914-3942.  doi: 10.1016/j.jde.2020.03.017.  Google Scholar

[5]

C. Cowan and A. Razani, Singular solutions of a Lane-Emden system, Discrete Contin. Dyn. Syst., 41 (2021), 621-656.  doi: 10.3934/dcds.2020291.  Google Scholar

[6]

J. DávilaM. del Pino and M. Musso, The Supercritical Lane-Emden-Fowler Equation in Exterior Domains, Commun. Partial Differ. Equ., 32 (2007), 1225-1243.  doi: 10.1080/03605300600854209.  Google Scholar

[7]

J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680.  doi: 10.1142/S0219199707002575.  Google Scholar

[8]

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.  Google Scholar

[9]

M. del PinoP. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron problem, Calc. Var. Partial Differ. Equ., 16 (2003), 113-145.  doi: 10.1007/s005260100142.  Google Scholar

[10]

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.  Google Scholar

[11]

M. Gherga and V. Rădulescu, Nonlinear PDEs, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-22664-9.  Google Scholar

[12]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Commun. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.  Google Scholar

[13]

N. GrenonF. Murat and A. Porretta, Existence and a priori estimate for elliptic problems with subquadratic gradient dependent terms, C. R. Acad. Sci. Paris, Ser. I, 342 (2006), 23-28.  doi: 10.1016/j.crma.2005.09.027.  Google Scholar

[14]

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.  Google Scholar

[15]

R. Mazzeo and F. Pacard, A construction of singular solutions for a semilinear elliptic equation using asymptotic analysis, J. Differ. Geom., 44 (1996), 331-370.   Google Scholar

[16]

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.  Google Scholar

[17]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.   Google Scholar

[18]

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.  Google Scholar

[19]

M. Struwe, Variational Methods–Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Berlin, Springer-Verlag, 1990. doi: 10.1007/978-3-662-02624-3.  Google Scholar

[20]

Z. Zhang, Boundary blow-up elliptic problems with nonlinear gradient terms, J. Differ. Equ., 228 (2006), 661-684.  doi: 10.1016/j.jde.2006.02.003.  Google Scholar

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