Multiplicity of solutions for the plasma problem in two dimensions

Let Ω   be a bounded domain in R2R2, u+=uu+=u if u⩾0u⩾0, u+=0u+=0 if u<0u<0, u−=u+−uu−=u+−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma(Pλ){Δu−λu−=0,inΩ,u=c,on∂Ω,∫∂Ω∂u∂νds=I, where ν is the outward unit normal of ∂Ω at x, c is a constant which is unprescribed, and I   is a given positive constant. The set Ωp={x∈Ω,u(x)<0} is called plasma set. Existence of solutions whose plasma set consisting of one component and asymptotic behavior of plasma set were studied by Caffarelli and Friedman (1980) [3] for large λ. Under the condition that the homology of Ω   is nontrivial we obtain in this paper by a constructive way that for any given integer k⩾1k⩾1, there is λk>0λk>0 such that for λ>λkλ>λk, (PλPλ) has a solution with plasma set consisting of k components.