Steep potential well may help Kirchhoff type equations to generate multiple solutions. (January 2020) - Sun, Juntao; Wu, Tsung-fang
Abstract: In this paper, we consider a nonlinear Kirchhoff type problem with steep potential well: − a ∫ R 3 ∇ u 2 d x + b Δ u + λ V ( x ) u = f x | u | p − 2 u in R 3, u ∈ H 1 ( R 3 ), where a, b, λ > 0, 2 < p < 4, V ∈ C ( R 3, R + ) and f ∈ L ∞ ( R 3, R ) . Such problem cannot be studied by applying variational methods in a standard way, even by restricting its corresponding energy functional on the Nehari manifold, because Palais–Smale sequences may not be bounded. In this paper, we introduce a novel constraint method to prove the existence of one and two positive solutions under the different assumptions on V, respectively. We conclude that steep potential well V may help Kirchhoff type equations to generate multiple solutions, which has never been involved before.