A bifurcation theory for some nonlinear elliptic equations

• Autorzy: Biagio Ricceri
• Języki publikacji: EN

Abstrakty

EN

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 ($P_{λ}$).

Kategorie tematyczne

• 35J20: Variational methods for second-order elliptic equations
• 35B32: Bifurcation

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2003
• Tom: 95
• Numer: 1
• Strony: 139-151

Daty

wydano

2003

Twórcy

autor

Biagio Ricceri

• Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Identyfikatory

DOI

10.4064/cm95-1-12