The Neumann Problem for a Degenerate Elliptic System Near Resonance
Keyword(s):
This paper studies the following system of degenerate equations-divpx∇u+qxu=αu+βv+g1x,v+h1x,x∈Ω,-div(p(x)∇v)+q(x)v=βu+αv+g2(x,u)+h2(x),x∈Ω,∂u/∂ν=∂v/∂ν=0,x∈∂Ω.HereΩ⊂Rnis a boundedC2domain, andνis the exterior normal vector on∂Ω. The coefficient functionpmay vanish inΩ¯,q∈Lr(Ω)withr>ns/(2s-n), s>n/2. We show that the eigenvalues of the operator-div(p(x)∇u)+q(x)uare discrete. Secondly, when the linear part is near resonance, we prove the existence of at least two different solutions for the above degenerate system, under suitable conditions onh1,h2,g1, andg2.
2002 ◽
Vol 1
(1)
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pp. 127-134
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Keyword(s):
2017 ◽
Vol 8
(1)
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pp. 661-678
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