A bifurcation theory for some nonlinear elliptic equations

Biagio 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 -