Nodal solutions of a perturbed elliptic problem

This paper is devoted to the perturbed elliptic boundary value problem {−Δuu=f(x,u)+ϵg(x,u)=0in Ω,on ∂Ω,(1) where Ω is a bounded domain in ℝN with smooth boundary ∂Ω, Δ is the Laplacian operator, ϵ is a parameter, and f and g belong to C(Ω⎯⎯⎯⎯⎯×ℝ). The authors prove two results concerning the existence of multiple nodal solutions to (1). The first theorem states that if f and g belong to C1(Ω⎯⎯⎯⎯⎯×ℝ), f is superlinear and odd in the second variable, and |f′t| is assumed to satisfy a certain asymptotic condition at infinity, then for any j∈ℕ there exists ϵj\u3e0 such that if |ϵ|≤ϵj then the problem (1) possesses at least j distinct nodal solutions corresponding to positive critical values. The second theorem shows a similar consequence for a continuous sublinear function f and for any continuous function g. The proofs rely on the notion of essential value introduced by M. Degiovanni and S. Lancelotti [Differential Integral Equations 8 (1995), no. 5, 981–992; MR1325542]. The results in this paper improve previous results of S. J. Li and Z. L. Liu [J. Differential Equations 185 (2002), no. 1, 271–280; MR1935639] in which the nodal properties of the solutions were not considered