Characterization for the existence of bounded solutions to elliptic equations

Adnan Ben Aziza; Mohamed Ben Chrouda

doi:10.3934/dcds.2019049

Article Contents

2019, Volume 39, Issue 2: 1157-1170. Doi: 10.3934/dcds.2019049

This issue Previous Article Extremal functions for an embedding from some anisotropic space, involving the "one Laplacian" Next Article Finite-time blowup for a Schrödinger equation with nonlinear source term

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

Received: April 30, 2018

Revised: July 31, 2018

Published: January 2019

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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:

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

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	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.
	D. H. Armitage and S. J. Gardiner, Classical Potential Theory, Springer, London, 2001. doi: 10.1007/978-1-4471-0233-5.
	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.
	J. Bliedtner and W. Hansen, Potential Theory, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-71131-2.
	K. L. Chung, Lectures from Markov Processes to Brownian Motion, Springer, Verlag, Berlin, 1982.
	Ph. Clément and G. Sweers , Getting a solution between sub and supersolutions without monotone iteration, Rend. Istit. Mat. Univ. Trieste, 19 (1987) , 189-194.
	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.
	E. B. Dynkin, Diffusions, Superdiffusions and Partial Differential Equations, American Math. Soc, Providence, Rhode Island, Colloquium Publications, 2002. doi: 10.1090/coll/050.
	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.
	N. Kawano , On bounded entire solutions of semilinear elliptic equations, Hiroshima Math. J, 14 (1984) , 125-158.
	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.
	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.
	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.
	M. Marcus and L. Véron, Nonlinear Second Order Elliptic Equations Involving Measures, De Gruyter Series in Nonlinear Analysis and Applications, 21 2014.
	M. Naito , A note on bounded positive entire solutions of semilinear elliptic equations, Hiroshima Math. J, 14 (1984) , 211-214.
	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.
	S. C. Port and C. J. Stone, Brownian Motion and Classical Potential Theory, Academic Press, New York-London, 1978.
	D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture notes in Mathematics 309, Springer-Verlag, Berlin/Heidelberg/New york, 1973.
	M. Sharpe, General Theory of Markov Processes, Academic Press, Boston, 1988.
	H. Yamabe , On a deformation of Riemannian structures on compact manifolds, Osaka Math. J, 12 (1960) , 21-37.
	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.