On Schrödinger–Poisson systems under the effect of steep potential well (2 < p < 4)

Abstract

We investigate the existence of nontrivial solutions for a class of Schrödinger–Poisson systems with steep potential well V and the nonlinearity a(x)|u|p−2u(2 < p < 4) in R3. Such a 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, by developing a novel constraint method, we find one positive solution under suitable assumptions when 2 < p < 4. In particular, one positive ground state solution is found when 1+733≤p<4. An interesting point is that unlike some methods that only work for the case of either 2 < p ≤ 3 or 3 < p < 4 {see the work by Jiang and Zhou [J. Differ. Equations 251, 582–608 (2011)]; Sun et al. [Z. Angew. Math. Phys. 68, 73 (2017)]; and Zhao et al. [J. Differ. Equations 255, 1–23 (2013)]}, our method is applicable to all 2 < p < 4.