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

Singular solutions of a Hénon equation involving a nonlinear gradient term

Craig Cowan 1, and  Abdolrahman Razani 2,,

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.
Keywords: Hénon equation, singular solutions, Laplace equation involving the gradient term, semilinear elliptic equations, perturbations of nonlinear operators.
Mathematics Subject Classification: Primary: 35J61, 35J75, 35J91; Secondary: 47H14.
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. Aghajani, C. 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. Aghajani, C. 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. Dávila, M. 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 Pino, P. 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. Grenon, F. 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. Aghajani, C. 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. Aghajani, C. 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. Dávila, M. 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 Pino, P. 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. Grenon, F. 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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