We study the following nonlinear elliptic system of Lane–Emden type $−Δu = sgn(v)|v|^{p−1}$ in Ω, $−Δv = −λ sgn(u)|u|^{1/{p−1}} + f(x, u)$ in Ω, u = v = 0 on ∂Ω, where λ ∈ R. If λ ≥ 0 and Ω is an unbounded cylinder, i.e., Ω = Ω ×$R^{N−m} ⊂ R^N$ , N − m ≥ 2, m ≥ 1, existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if λ ∈ R and Ω is a bounded domain in $R^N$ , N ≥ 3. In particular, a good finite dimensional decomposition of the Banach space in which we work is given.
Existence and multiplicity results for some elliptic systems in unbounded cylinders
SALVATORE, Addolorata
2013-01-01
Abstract
We study the following nonlinear elliptic system of Lane–Emden type $−Δu = sgn(v)|v|^{p−1}$ in Ω, $−Δv = −λ sgn(u)|u|^{1/{p−1}} + f(x, u)$ in Ω, u = v = 0 on ∂Ω, where λ ∈ R. If λ ≥ 0 and Ω is an unbounded cylinder, i.e., Ω = Ω ×$R^{N−m} ⊂ R^N$ , N − m ≥ 2, m ≥ 1, existence and multiplicity results are proved by means of the Principle of Symmetric Criticality and some compact imbeddings in partially spherically symmetric spaces. We are able to state existence and multiplicity results also if λ ∈ R and Ω is a bounded domain in $R^N$ , N ≥ 3. In particular, a good finite dimensional decomposition of the Banach space in which we work is given.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
