It is proved that the elliptic problem -Δu=f(x,u) in Ω,u=0 on ∂Ωon a bounded domain Ω⊂RN with smooth boundary has two sign-changing solutions and one positive solution if f∈C1(Ω̄ ×R,R) satisfies μi<fu′(x,0)<μi+1 for some i≥2, limsupu→+∞f(x,u)/u<μ1<liminfu→-∞f(x,u)/u, and limsupu→-∞f(x,u)/u<∞, where μ1<μ2≤μ3≤ ⋯ are eigenvalues of -Δ with 0-Dirichlet boundary condition on Ω counting with their multiplicity. © 2005 Elsevier Ltd. All rights reserved.
Liu, Z., & Sun, J. (2005). An elliptic problem with jumping nonlinearities. Nonlinear Analysis, Theory, Methods and Applications, 63(8), 1070–1082. https://doi.org/10.1016/j.na.2005.03.109