Minimax characterization of solutions for a semi-linear elliptic equation with lack of compactness

Abstract In this paper conditions ensuring bifurcation from any boundary point of the spectrum is studied for a class of nonlinear operators. We give a general minimax result which allows an enlargement of the class of non-linearities which has been studied up to now. The general result is applied to study the existence of solutions (u, λ) ∈ H1 (ℝN) × ℝ for the equation − Δ u + p u − N ( u ) = λ u , u ≠ 0 , where λ is located in a prescribed gap of the spectrum of −∆u + pu. The function p is periodic and the superlinear term N derives from a potential but is not assumed to be compact.