A semilinear elliptic eigenvalue problem, I
Grant Keady, John Norbury · Proceedings of the Royal Society of Edinburgh Section A Mathematics · 1980
Synopsis The nonlinear operator equation (N), , describes both semilinear elliptic boundary value problems (P) and their natural discretisations (P h ). (Here is a positive compact linear operator and [ t ] + ≡ ( t + | t |)/2 for all t ∊ ℝ.) It is proved that, for q ≧ 0 ( q ≢ 0), in a Banach lattice E the equation (N) has an unbounded continuum ℱ of nontrivial solutions (λ, ψ) ∊ ℝ × E bifurcating from infinity at (λ 1 , ∞) (here is the first (positive) eigenvalue of ). All nontrivial solutions (λ, ψ) have λ ≧ λ 1 , and if maps a smaller cone K s , into itself then there is a ℱ s ⊂ℝ× K s with similar properties to ℱ. The existence theory for (N) is applied to problems (P) and (P h ) which are defined on the simply-connected domain Ω. It is shown that the projection of ℱ on the λ-axis is either unbounded if the continuous function q > 0 except on a set of measure zero in Ω, or is bounded if q ≡ 0 on a subdomain of Ω. If is the second eigenvalue for in (P h ) then there is at most one nontrivial solution for each λ satisfying λ 1 < λ < λ 2 ; in the corresponding uniqueness result for (P) q is restricted to being strictly positive somewhere on ∂Ω Additional properties for the solutions of (P) and (P h ) are also proved.