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2007 Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains
Tsung-Fang Wu
Abstr. Appl. Anal. 2007: 1-25 (2007). DOI: 10.1155/2007/18187

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 ) .

Citation

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Tsung-Fang Wu. "Existence and Multiplicity of Positive Solutions for Dirichlet Problems in Unbounded Domains." Abstr. Appl. Anal. 2007 1 - 25, 2007. https://doi.org/10.1155/2007/18187

Information

Published: 2007
First available in Project Euclid: 27 February 2008

zbMATH: 1157.35377
MathSciNet: MR2302187
Digital Object Identifier: 10.1155/2007/18187

Rights: Copyright © 2007 Hindawi

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