Abstract
We will show that under suitable conditions onfandh, there exists a positive numberλ∗such that the nonhomogeneous elliptic equation−Δu+u=λ(f(x,u)+h(x))inΩ,u∈H01(Ω),N≥2, has at least two positive solutions ifλ∈(0,λ∗), a unique positive solution ifλ=λ∗, and no positive solution ifλ>λ∗, whereΩis the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.
Highlights
Let 2∗ = 2N/(N − 2) for N ≥ 3, 2∗ = ∞ for N = 2
We study the existence, nonexistence, and multiplicity of solutions of the equation
−Δu + u = λ f (x, u) + h(x) in Ω, u in H01(Ω), u > 0 in Ω, N ≥ 2, (1.1)λ where λ > 0, N = m + n ≥ 2, n ≥ 1, 0 ∈ ω ⊆ Rm is a smooth bounded domain, S = ω × Rn, D is a smooth bounded domain in RN such that D ⊂⊂ S, Ω = S\D is the exterior of this domain in the strip
Summary
Let 2∗ = 2N/(N − 2) for N ≥ 3, 2∗ = ∞ for N = 2. We study the existence, nonexistence, and multiplicity of solutions of the equation. (1.1)λ where λ > 0, N = m + n ≥ 2, n ≥ 1, 0 ∈ ω ⊆ Rm is a smooth bounded domain, S = ω × Rn, D is a smooth bounded domain in RN such that D ⊂⊂ S, Ω = S\D is the exterior of this domain in the strip. Associated to (1.1)λ, we consider the functional I, for u ∈ H01(Ω), I(u) = 1 |∇u|2 + u2 dx − λ F x, u+ dx − λ h(x)u dx, 2Ω
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