Abstract
In this paper, we consider a nonlinear Kirchhoff type problem with steep potential well: −a∫R3∇u2dx+bΔu+λV(x)u=fx|u|p−2uin R3,u∈H1(R3),where a,b,λ>0, 2<p<4, V∈C(R3,R+) and f∈L∞(R3,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.
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