Nodal Solutions of a Perturbed Elliptic Problem

Multiple nodal solutions are obtained for the elliptic problem

−Δu=f(x,u) + εg (x,u) in Ω,

u= 0 on ∂Ω,

where ε is a parameter, Ω is a smooth bounded domain in R ^N , f∈C(Ω ^¯ ×R), and g∈C(Ω ^¯ ×R). For a superlinear C ¹ function f which is odd in u and for any C ¹ function g, we prove that for any j∈N there exists ε _j >0 such that if |ε|≤ε _j then the above problem possesses at least j distinct nodal solutions. Except C ¹ continuity no further condition is needed for g. We also prove a similar result for a continuous sublinear function f and for any continuous function g. Results obtained here refine earlier results of S. J. Li and Z. L. Liu in which the nodal property of the solutions was not considered.