The effect of the domain topology on the number of positive solutions of an elliptic Kirchhoff problem. (April 2016) - Santos Junior, João R.
Abstract: Using minimax methods and Lusternik–Schnirelmann theory, we study multiple positive solutions for the Schrödinger–Kirchhoff equation M ( ∫ Ω λ | ∇ u | 2 d x + ∫ Ω λ u 2 d x ) [ − Δ u + u ] = f ( u ) in Ω λ = λ Ω . The set Ω ⊂ R 3 is a smooth bounded domain, λ > 0 is a parameter, M is a general continuous function and f is a superlinear continuous function with subcritical growth. Our main result relates, for large values of λ, the number of solutions with the least number of closed and contractible in Ω ¯ which cover Ω ¯ .