A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri

Colloquium Mathematicae, Tome 96 (2003), p. 139-151 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ($P_{λ}$).

Détail

Publié le : 2003-01-01

Zbl 1082.35026

EUDML-ID : urn:eudml:doc:285274