Characterization for the existence of bounded solutions to elliptic equations

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February 2019, 39(2): 1157-1170. doi: 10.3934/dcds.2019049

Characterization for the existence of bounded solutions to elliptic equations

Adnan Ben Aziza ^1, and Mohamed Ben Chrouda ^2,,

1.	LAMMDA-ESST Hammam Sousse, Université de Sousse, Tunisie
2.	LAMMDA-ISIM Monastir, Université de Monastir, Tunisie

* Corresponding author: mohamed.benchrouda@isimm.rnu.tn

Received May 2018 Revised August 2018 Published November 2018

We give necessary and sufficient conditions for which the elliptic equation

$\Delta u = \rho (x)\Phi (u)\;\;\;\;{\rm{in}}\;\;\;\;{\mathbb{R}^d}\;\;\;(d \ge 3)$

has nontrivial bounded solutions.

Keywords: Elliptic equation, bounded solution, thinness at ∞, Green function.

Mathematics Subject Classification: Primary: 31B05, 35B08; Secondary: 35J08, 35J91.

Citation: Adnan Ben Aziza, Mohamed Ben Chrouda. Characterization for the existence of bounded solutions to elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1157-1170. doi: 10.3934/dcds.2019049

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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