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Quantitative analysis of a system of integral equations with weight on the upper half space
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 |
$ \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*} $ |
$ \Omega $ |
$ \mathbb{R}^N $ |
$ \alpha $ |
$ p $ |
$ B_1 $ |
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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