The Number of Positive Solutions for Schrödinger–Poisson System under the Effect of Eigenvalue

We investigate the eigenvalue problem for the Schrödinger–Poisson system as follows: where is nonnegative and is a potential well with bottom A bounded Cerami sequence is established by using the truncation technique, decomposition of Hilbert space and mountain pass theorem. Then exploiting the property of steep potential well, under some proper assumptions on , we conclude that one positive solution exists for while two positive solutions exist for when where is the principal eigenvalue of in with weight function and is the corresponding principal eigenfunction. Moreover, an interesting phenomenon is that when , one positive solution is always present near for unaffected by the non‐local term, however large its norm might be.