Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains

Tsung-Fang Wu

Abstract and Applied Analysis, 2007, vol. 2007, 1-21

Abstract:

We consider the elliptic problem − Δ u + u = b ( x ) | u | p − 2 u + h ( x ) in Ω , u ∈ H 0 1 ( Ω ) , where 2 < p < ( 2 N / ( N − 2 ) )  ( N ≥ 3 ) ,  2 < p < ∞  ( N = 2 ) ,  Ω is a smooth unbounded domain in ℠N ,  b ( x ) ∈ C ( Ω ) , and h ( x ) ∈ H − 1 ( Ω ) . We use the shape of domain Ω to prove that the above elliptic problem has a ground-state solution if the coefficient b ( x ) satisfies b ( x ) → b ∞ > 0 as | x | → ∞ and b ( x ) ≥ c for some suitable constants c ∈ ( 0 , b ∞ ) , and h ( x ) ≡ 0 . Furthermore, we prove that the above elliptic problem has multiple positive solutions if the coefficient b ( x ) also satisfies the above conditions, h ( x ) ≥ 0 and 0 < ‖ h ‖ H − 1 < ( p − 2 ) ( 1 / ( p − 1 ) ) ( p − 1 ) / ( p − 2 ) [ b sup S p ( Ω ) ] 1 / ( 2 − p ) , where S ( Ω ) is the best Sobolev constant of subcritical operator in H 0 1 ( Ω ) and b sup = sup x ∈ Ω b ( x ) .

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnlaaa:018187

DOI: 10.1155/2007/18187

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