This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: -aij(x)∂iju(x) - q i(x)∂iu(x)=f(x,u(x)), x ∈ ℝN. The aim is to prove uniqueness of positive- bounded solutions-Liouville-type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the linearized operator, which guarantees the existence result as well as the Liouville property. Then, following another approach, we establish other results relying on the sign of the principal eigenvalue of the linearized operator about u= 0, of some limit operator at infinity which we define here. This framework will be seen to be the most general one. We also derive the large time behavior for the associated evolution equation. © Springer-Verlag 2007.

Liouville-type results for semilinear elliptic equations in unbounded domains / Berestycki, H.; Hamel, F.; Rossi, L.. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 186:3(2007), pp. 469-507. [10.1007/s10231-006-0015-0]

Liouville-type results for semilinear elliptic equations in unbounded domains

Rossi L.
2007

Abstract

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: -aij(x)∂iju(x) - q i(x)∂iu(x)=f(x,u(x)), x ∈ ℝN. The aim is to prove uniqueness of positive- bounded solutions-Liouville-type theorems. Along the way, we establish also various existence results. We first derive a sufficient condition, directly expressed in terms of the coefficients of the linearized operator, which guarantees the existence result as well as the Liouville property. Then, following another approach, we establish other results relying on the sign of the principal eigenvalue of the linearized operator about u= 0, of some limit operator at infinity which we define here. This framework will be seen to be the most general one. We also derive the large time behavior for the associated evolution equation. © Springer-Verlag 2007.
2007
Linear and semi-linear elliptic equations; Liouville-type results; maximum principles; periodic and almost-periodic equations; principal eigenvalues
01 Pubblicazione su rivista::01a Articolo in rivista
Liouville-type results for semilinear elliptic equations in unbounded domains / Berestycki, H.; Hamel, F.; Rossi, L.. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 186:3(2007), pp. 469-507. [10.1007/s10231-006-0015-0]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1464011
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