Invariant criteria for existence of bounded positive solutions

Dong Ye; Feng Zhou

doi:10.3934/dcds.2005.12.413

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2005, Volume 12, Issue 3: 413-424. Doi: 10.3934/dcds.2005.12.413

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Invariant criteria for existence of bounded positive solutions

Dong Ye^1, and
Feng Zhou^2,

1.
Département de Mathématiques, Université de Cergy-Pontoise, Site Saint-Martin, BP 222, 95302 Cergy-Pontoise Cedex
2.
Department of Mathematics, East China Normal University, Shanghai 200062

Received: September 2003

Revised: July 2004

Published: April 2005

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Abstract

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.

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Mathematics Subject Classification: 35J60, 35B05, 35B50, 35B35.

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