The number of positive solutions affected by the weight function to Kirchhoff type equations in high dimensions. (July 2020) - Zhang, Jian; Sun, Juntao; Wu, Tsung-fang
Abstract: We study the multiplicity of positive solutions for the following Kirchhoff type equations: − ε 2 M ε 2 − N ∇ u L 2 2 Δ u + u = Q x u p − 2 u in R N, u ∈ H 1 ( R N ), where N ≥ 4, ε > 0 is a small parameter, M t = a t + b a, b > 0 and 2 < p < 2 ∗ = 2 N N − 2 . We assume that the weight function Q ∈ C ( R N, R + ) has k maximum points in R N . By using a novel constraint approach as well as the barycenter map, we show that the above problem admits k positive solutions when N = 4 while 2 k positive solutions when N ≥ 5 for ε, a sufficiently small.