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Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian"
Characterization for the existence of bounded solutions to elliptic equations
1. | LAMMDA-ESST Hammam Sousse, Université de Sousse, Tunisie |
2. | LAMMDA-ISIM Monastir, Université de Monastir, Tunisie |
$\Delta u = \rho (x)\Phi (u)\;\;\;\;{\rm{in}}\;\;\;\;{\mathbb{R}^d}\;\;\;(d \ge 3)$ |
References:
[1] |
R. Alsaedi, H. Mâagli, V. D. Radulescu and N. Zeddini,
Entire bounded solutions versus fixed points for nonlinear elliptic equations with indefinite weight, Fixed Point Theory, 17 (2016), 255-265.
|
[2] |
D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer, London, 2001.
doi: 10.1007/978-1-4471-0233-5. |
[3] |
M. Ben Chrouda and M. Ben Fredj,
Nonnegative entire bounded solutions to some semilinear equations involving the fractional laplacian, Potential Anal, 48 (2018), 495-513.
doi: 10.1007/s11118-017-9645-7. |
[4] |
J. Bliedtner and W. Hansen, Potential Theory, Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-642-71131-2. |
[5] |
K. L. Chung, Lectures from Markov Processes to Brownian Motion, Springer, Verlag, Berlin, 1982. |
[6] |
Ph. Clément and G. Sweers,
Getting a solution between sub and supersolutions without monotone iteration, Rend. Istit. Mat. Univ. Trieste, 19 (1987), 189-194.
|
[7] |
L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault,
Entire large solutions for semilinear elliptic equations, J. Differential Equations, 253 (2012), 2224-2251.
doi: 10.1016/j.jde.2012.05.024. |
[8] |
E. B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, American Math. Soc, Providence, Rhode Island, Colloquium Publications, 2002.
doi: 10.1090/coll/050. |
[9] |
K. El Mabrouk,
Entire bounded solutions for a class of sublinear elliptic equations, Nonlinear Anal, 58 (2004), 205-218.
doi: 10.1016/j.na.2004.01.004. |
[10] |
N. Kawano,
On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J, 14 (1984), 125-158.
|
[11] |
O. D. Kellogg, Foundations of Potential Theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31 Springer-Verlag, Berlin-New York, 1967. |
[12] |
A. V. Lair and A. W. Wood,
Large solutions of sublinear elliptic equations, Nonlinear Anal, 39 (2000), 745-753.
doi: 10.1016/S0362-546X(98)00233-8. |
[13] |
J. F. Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations, Originally published by Birkhliuser Verlag, 1999.
doi: 10.1007/978-3-0348-8683-3. |
[14] |
M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter Series in Nonlinear Analysis and Applications, 21 2014. |
[15] |
M. Naito,
A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J, 14 (1984), 211-214.
|
[16] |
W. M. Ni,
On the elliptic equation $Δ u + K(x) u^{\frac{n+2}{n-2}}=0$, its generalizations and applications in geometry, Indiana Univ. Math. J, 31 (1982), 493-529.
doi: 10.1512/iumj.1982.31.31040. |
[17] |
S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Academic Press, New York-London, 1978. |
[18] |
D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture notes in Mathematics 309, Springer-Verlag, Berlin/Heidelberg/New york, 1973. |
[19] |
M. Sharpe, General Theory of Markov Processes, Academic Press, Boston, 1988. |
[20] |
H. Yamabe,
On a deformation of Riemannian structures on compact manifolds, Osaka Math. J, 12 (1960), 21-37.
|
[21] |
D. Ye and F. Zhou,
Invariant criteria for existence of bounded positive solutions, Discrete Contin. Dynam. Systems, 12 (2005), 413-424.
doi: 10.3934/dcds.2005.12.413. |
show all references
References:
[1] |
R. Alsaedi, H. Mâagli, V. D. Radulescu and N. Zeddini,
Entire bounded solutions versus fixed points for nonlinear elliptic equations with indefinite weight, Fixed Point Theory, 17 (2016), 255-265.
|
[2] |
D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer, London, 2001.
doi: 10.1007/978-1-4471-0233-5. |
[3] |
M. Ben Chrouda and M. Ben Fredj,
Nonnegative entire bounded solutions to some semilinear equations involving the fractional laplacian, Potential Anal, 48 (2018), 495-513.
doi: 10.1007/s11118-017-9645-7. |
[4] |
J. Bliedtner and W. Hansen, Potential Theory, Springer-Verlag, Berlin, 1986.
doi: 10.1007/978-3-642-71131-2. |
[5] |
K. L. Chung, Lectures from Markov Processes to Brownian Motion, Springer, Verlag, Berlin, 1982. |
[6] |
Ph. Clément and G. Sweers,
Getting a solution between sub and supersolutions without monotone iteration, Rend. Istit. Mat. Univ. Trieste, 19 (1987), 189-194.
|
[7] |
L. Dupaigne, M. Ghergu, O. Goubet and G. Warnault,
Entire large solutions for semilinear elliptic equations, J. Differential Equations, 253 (2012), 2224-2251.
doi: 10.1016/j.jde.2012.05.024. |
[8] |
E. B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, American Math. Soc, Providence, Rhode Island, Colloquium Publications, 2002.
doi: 10.1090/coll/050. |
[9] |
K. El Mabrouk,
Entire bounded solutions for a class of sublinear elliptic equations, Nonlinear Anal, 58 (2004), 205-218.
doi: 10.1016/j.na.2004.01.004. |
[10] |
N. Kawano,
On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J, 14 (1984), 125-158.
|
[11] |
O. D. Kellogg, Foundations of Potential Theory, Reprint from the first edition of 1929. Die Grundlehren der Mathematischen Wissenschaften, Band 31 Springer-Verlag, Berlin-New York, 1967. |
[12] |
A. V. Lair and A. W. Wood,
Large solutions of sublinear elliptic equations, Nonlinear Anal, 39 (2000), 745-753.
doi: 10.1016/S0362-546X(98)00233-8. |
[13] |
J. F. Le Gall, Spatial Branching Processes, Random Snakes and Partial Differential Equations, Originally published by Birkhliuser Verlag, 1999.
doi: 10.1007/978-3-0348-8683-3. |
[14] |
M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter Series in Nonlinear Analysis and Applications, 21 2014. |
[15] |
M. Naito,
A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J, 14 (1984), 211-214.
|
[16] |
W. M. Ni,
On the elliptic equation $Δ u + K(x) u^{\frac{n+2}{n-2}}=0$, its generalizations and applications in geometry, Indiana Univ. Math. J, 31 (1982), 493-529.
doi: 10.1512/iumj.1982.31.31040. |
[17] |
S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Academic Press, New York-London, 1978. |
[18] |
D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture notes in Mathematics 309, Springer-Verlag, Berlin/Heidelberg/New york, 1973. |
[19] |
M. Sharpe, General Theory of Markov Processes, Academic Press, Boston, 1988. |
[20] |
H. Yamabe,
On a deformation of Riemannian structures on compact manifolds, Osaka Math. J, 12 (1960), 21-37.
|
[21] |
D. Ye and F. Zhou,
Invariant criteria for existence of bounded positive solutions, Discrete Contin. Dynam. Systems, 12 (2005), 413-424.
doi: 10.3934/dcds.2005.12.413. |
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