April  2005, 12(3): 413-424. doi: 10.3934/dcds.2005.12.413

Invariant criteria for existence of bounded positive solutions

1. 

Département de Mathématiques, Université de Cergy-Pontoise, Site Saint-Martin, BP 222, 95302 Cergy-Pontoise Cedex, France

2. 

Department of Mathematics, East China Normal University, Shanghai 200062, China

Received  September 2003 Revised  July 2004 Published  December 2004

We consider semilinear elliptic equations $\Delta u \pm \rho(x)f(u) = 0$, or more generally $\Delta u + \varphi(x, u) = 0$, posed in $\R^N$ ($N\geq 3$). We prove that the existence of entire bounded positive solutions is closely related to the existence of bounded solution for $\Delta u + \rho(x) = 0$ in $\mathbb R^N$. Many sufficient conditions which are invariant under the isometry group of $\mathbb R^N$ are established. Our proofs use the standard barrier method, but our results extend many earlier works in this direction. Our ideas can also be applied for the existence of large solutions, for the exterior domain problem and for the system situations.
Citation: Dong Ye, Feng Zhou. Invariant criteria for existence of bounded positive solutions. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 413-424. doi: 10.3934/dcds.2005.12.413
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