The fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains

Abstract

In this article, we study a p-Laplacian Schrödinger-Poisson system involving a parameter q ≠ 0 q\ne 0 in bounded domains. By using the Nehari manifold and the fibering method, we obtain the non-existence and multiplicity of nontrivial solutions. On one hand, there exists q * > 0 {q}^{* }\gt 0 such that only the trivial solution is admitted for q ∈ ( q * , + ∞ ) . q\in \left({q}^{* },+\infty ). On the other hand, there are two positive solutions existing for q ∈ ( 0 , q 0 * + ε ) q\in \left(0,{q}_{0}^{* }+\varepsilon ) , where ε > 0 \varepsilon \gt 0 and q 0 * + ε < q * . {q}_{0}^{* }+\varepsilon \lt {q}^{* }. In particular, q * {q}^{* } and q 0 * {q}_{0}^{* } correspond to the supremum for the nonlinear generalized Rayleigh quotients, respectively. The specific form of the nonlinear generalized Rayleigh quotients is calculated. Moreover, it is worth mentioning that we also obtain the qualitative properties associated with the energy level of the solutions.