Slawomir M. Rybicki

Nicolaus Copernicus University (Poland)

Global bifurcations of solutions of elliptic systems

The aim of my talk is to present Rabinowitz alternative for systems of elliptic differential equations of the form

- Δu = ∇uF (u,λ)  in Ω,
   u = 0          on ∂Ω,

where

  1. Ω RN is an open, bounded subset of RN, with boundary of the class C1-,
  2. F ∈ C2(Rm × R,R),
  3. dsF(x,λ) = λ2Ax,x+ η(x,λ), where
    1. A is a symmetric (m × m)-matrix,
    2. xη(0) = 0, for any λ ∈ R,
    3. x2η(0) = 0, for any λ ∈ R,
  4. there are C > 0 and 1 p < (N + 2)(N - 2)-1 such that for any (x,λ) ∈ Rn ×R the following inequality holds true |∇xF(x,λ)|≤ C(1+ |x|p).