Abstract
We study the existence and multiplicity of nontrivial solutions for a Schrödinger–Poisson system involving critical nonlocal term and general nonlinearity. Based on the variational method and analysis technique, we obtain the existence of two nontrivial solutions for this system.
Highlights
Introduction and Main ResultThe Schrödinger–Poisson system is usually used to describe solitary waves for the nonlinear stationary Schrödinger equations interacting with an electromagnetic field
For the case q = 1, gðx, uÞ = λf ðxÞ, λ > 0 is a real number, f ≥ 0, and f ∈ L2∗/ð2∗−1ÞðRN Þ, in [13]; when η = 1, authors proved that system (1) has at least two positive solutions if 0 < λ < λ ∗ for some λ ∗ >0 small enough, and when η = −1, system (1) has at least one positive solution for any λ > 0
J ′ðuÞ = 0 and u is a nontrivial solution of problem (4)
Summary
Introduction and Main ResultThe Schrödinger–Poisson system is usually used to describe solitary waves for the nonlinear stationary Schrödinger equations interacting with an electromagnetic field. There are currently only a few results for the following Schrödinger–Poisson systems with critical nonlocal terms in a bounded domain [11,12,13]: 8 >>< For the case q = 1, gðx, uÞ = λf ðxÞ, λ > 0 is a real number, f ≥ 0, and f ∈ L2∗/ð2∗−1ÞðRN Þ, in [13]; when η = 1, authors proved that system (1) has at least two positive solutions if 0 < λ < λ ∗ for some λ ∗ >0 small enough, and when η = −1, system (1) has at least one positive solution for any λ > 0.
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