Multiplicity of solutions to nearly critical elliptic equation in the bounded domain of R3R3

We consider the following Dirichlet boundary value problemequation(0.1){−Δu=u5−ε+λuq,u>0in Ω;u=0on ∂Ω, where Ω   is a smooth bounded domain in R3R3, 10λ>0 and ε>0ε>0. By Lyapunov–Schmidt reduction method and the Mountain Pass Theorem, we prove that in suitable ranges for the parameters λ and ε, problem (0.1) has at least two solutions. Additionally if 2≤q<32≤q<3, we prove the existence of at least three solutions. Consequently, we prove a non-uniqueness result for a subcritical problem with an increasing nonlinearity.