The effect of the domain topology on the number of positive solutions of an elliptic Kirchhoff problem

Using minimax methods and Lusternik–Schnirelmann theory, we study multiple positive solutions for the Schrödinger–Kirchhoff equation M(∫Ωλ|∇u|2dx+∫Ωλu2dx)[−Δu+u]=f(u)M(∫Ωλ|∇u|2dx+∫Ωλu2dx)[−Δu+u]=f(u) in Ωλ=λΩΩλ=λΩ. The set Ω⊂R3Ω⊂R3 is a smooth bounded domain, λ>0λ>0 is a parameter, MM is a general continuous function and ff is a superlinear continuous function with subcritical growth. Our main result relates, for large values of λλ, the number of solutions with the least number of closed and contractible in Ω¯ which cover Ω¯.