# A bifurcation theory for some nonlinear elliptic equations

Colloquium Mathematicae (2003)

- Volume: 95, Issue: 1, page 139-151
- ISSN: 0010-1354

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topBiagio Ricceri. "A bifurcation theory for some nonlinear elliptic equations." Colloquium Mathematicae 95.1 (2003): 139-151. <http://eudml.org/doc/285274>.

@article{BiagioRicceri2003,

abstract = {We deal with the problem
⎧ -Δu = f(x,u) + λg(x,u), in Ω,
⎨ ($P_\{λ\}$)
⎩ $u_\{∣∂Ω\} = 0$
where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_\{λ\}$) admits a non-zero, non-negative strong solution $u_\{λ\} ∈ ⋂_\{p≥2\}W^\{2,p\}(Ω)$ such that $lim_\{λ→0⁺\} ||u_\{λ\}||_\{W^\{2,p\}(Ω)\} = 0$ for all p ≥ 2. Moreover, the function $λ ↦ I_\{λ\}(u_\{λ\})$ is negative and decreasing in ]0,λ*[, where $I_\{λ\}$ is the energy functional related to ($P_\{λ\}$).},

author = {Biagio Ricceri},

journal = {Colloquium Mathematicae},

language = {eng},

number = {1},

pages = {139-151},

title = {A bifurcation theory for some nonlinear elliptic equations},

url = {http://eudml.org/doc/285274},

volume = {95},

year = {2003},

}

TY - JOUR

AU - Biagio Ricceri

TI - A bifurcation theory for some nonlinear elliptic equations

JO - Colloquium Mathematicae

PY - 2003

VL - 95

IS - 1

SP - 139

EP - 151

AB - We deal with the problem
⎧ -Δu = f(x,u) + λg(x,u), in Ω,
⎨ ($P_{λ}$)
⎩ $u_{∣∂Ω} = 0$
where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ($P_{λ}$) admits a non-zero, non-negative strong solution $u_{λ} ∈ ⋂_{p≥2}W^{2,p}(Ω)$ such that $lim_{λ→0⁺} ||u_{λ}||_{W^{2,p}(Ω)} = 0$ for all p ≥ 2. Moreover, the function $λ ↦ I_{λ}(u_{λ})$ is negative and decreasing in ]0,λ*[, where $I_{λ}$ is the energy functional related to ($P_{λ}$).

LA - eng

UR - http://eudml.org/doc/285274

ER -

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