Singular solutions of a Hénon equation involving a nonlinear gradient term

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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 and 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.
[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.
[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.
[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.
[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.
[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.
[7]	J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680. doi: 10.1142/S0219199707002575.
[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.
[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.
[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.
[11]	M. Gherga and V. Rădulescu, Nonlinear PDEs, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-22664-9.
[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.
[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.
[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.
[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.
[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.
[17]	S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[7]	J. Dávila and L. Dupaigne, Perturbing singular solutions of the Gelfand problem, Commun. Contemp. Math., 9 (2007), 639-680. doi: 10.1142/S0219199707002575.
[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.
[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.
[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.
[11]	M. Gherga and V. Rădulescu, Nonlinear PDEs, Springer-Verlag, Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-22664-9.
[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.
[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.
[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.
[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.
[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.
[17]	S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Soviet. Math. Dokl., 6 (1965), 1408-1411.
[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.
[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.
[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.

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