Du, Yihong; Guo, Yuxia

Author(s)

Du, Yihong

Guo, Yuxia

Publication Date

2003

Abstract

We consider the elliptic problem -Δu - λu = a(x)g(u), with a(x) sign-changing and g(u) behaving like u^p, p > 1. Under suitable conditions on g(u) and a(x), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space ℝ^N. More precisely, we show that there exists Λ > 0 such that this equation on ℝ^N has no positive solution for λ > Λ, at least two positive solutions for λ ∈ (o,Λ), and at least one positive solution for λ ∈ (-∞,0]U{A}. Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.

Citation

Topological Methods in Nonlinear Analysis, 22(1), p. 69-92

ISSN

1230-3429

Link

https://hdl.handle.net/1959.11/3170

Publisher

Juliusz Schauder Center

Title

Mountain Pass Solutions and an Indefinite Superlinear Elliptic Problem on ℝ^N

Type of document

Journal Article

Entity Type

Publication

Author(s)	Du, Yihong Guo, Yuxia
Publication Date	2003
Abstract	We consider the elliptic problem -Δu - λu = a(x)g(u), with a(x) sign-changing and g(u) behaving like u^p, p > 1. Under suitable conditions on g(u) and a(x), we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space ℝ^N. More precisely, we show that there exists Λ > 0 such that this equation on ℝ^N has no positive solution for λ > Λ, at least two positive solutions for λ ∈ (o,Λ), and at least one positive solution for λ ∈ (-∞,0]U{A}. Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.
Citation	Topological Methods in Nonlinear Analysis, 22(1), p. 69-92
ISSN	1230-3429
Link	https://hdl.handle.net/1959.11/3170
Publisher	Juliusz Schauder Center
Title	Mountain Pass Solutions and an Indefinite Superlinear Elliptic Problem on ℝ^N
Type of document	Journal Article
Entity Type	Publication

Mountain Pass Solutions and an Indefinite Superlinear Elliptic Problem on ℝ^N

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