Exactly two positive solutions of nonhomogeneous semilinear elliptic equations in the half-space

Tsing San Hsu

doi:10.1016/j.na.2007.10.055

Exactly two positive solutions of nonhomogeneous semilinear elliptic equations in the half-space

Tsing San Hsu^*

^*Corresponding author for this work

Center for General Education

Research output: Contribution to journal › Journal Article › peer-review

3 Scopus citations

Abstract

In this article, we shall show that under suitable conditions on f and h, there exists a positive number λ^* such that the nonhomogeneous semilinear elliptic equation - Δ u + σ² u = λ (f (u) + h (x)) in R₊^N, u ∈ H₀¹ (R₊^N), N ≥ 3, has exactly two positive solutions if λ ∈ (0, λ^*), a unique positive solution if λ = λ^* and no positive solution if λ > λ^*. We also obtain some properties of the set of solutions.

Original language	English
Pages (from-to)	4324-4339
Number of pages	16
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	69
Issue number	12
DOIs	https://doi.org/10.1016/j.na.2007.10.055
State	Published - 15 12 2008

Keywords

Exactly two positive solutions
Nonhomogeneous
Semilinear elliptic equations
The half-space

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10.1016/j.na.2007.10.055

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abstract = "In this article, we shall show that under suitable conditions on f and h, there exists a positive number λ* such that the nonhomogeneous semilinear elliptic equation - Δ u + σ2 u = λ (f (u) + h (x)) in R+N, u ∈ H01 (R+N), N ≥ 3, has exactly two positive solutions if λ ∈ (0, λ*), a unique positive solution if λ = λ* and no positive solution if λ > λ*. We also obtain some properties of the set of solutions.",

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N2 - In this article, we shall show that under suitable conditions on f and h, there exists a positive number λ* such that the nonhomogeneous semilinear elliptic equation - Δ u + σ2 u = λ (f (u) + h (x)) in R+N, u ∈ H01 (R+N), N ≥ 3, has exactly two positive solutions if λ ∈ (0, λ*), a unique positive solution if λ = λ* and no positive solution if λ > λ*. We also obtain some properties of the set of solutions.

AB - In this article, we shall show that under suitable conditions on f and h, there exists a positive number λ* such that the nonhomogeneous semilinear elliptic equation - Δ u + σ2 u = λ (f (u) + h (x)) in R+N, u ∈ H01 (R+N), N ≥ 3, has exactly two positive solutions if λ ∈ (0, λ*), a unique positive solution if λ = λ* and no positive solution if λ > λ*. We also obtain some properties of the set of solutions.

KW - Exactly two positive solutions

KW - Nonhomogeneous

KW - Semilinear elliptic equations

KW - The half-space

UR - https://www.scopus.com/pages/publications/55049101327

U2 - 10.1016/j.na.2007.10.055

DO - 10.1016/j.na.2007.10.055

M3 - 文章

AN - SCOPUS:55049101327

SN - 0362-546X

VL - 69

SP - 4324

EP - 4339

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 12

ER -

Exactly two positive solutions of nonhomogeneous semilinear elliptic equations in the half-space

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Keywords

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