On a Schrödinger–Poisson system with singularity and critical nonlinearities

Abstract

In this paper, we study the Schrödinger-Poisson system with singularity and critical growth terms. By means of variational methods with an appropriate truncation argument, the existence and multiplicity of positive solutions are obtained.

Positive Ground State Solutions for Schrödinger-Poisson System with General Nonlinearity and Critical Exponent

“…To the best of our knowledge, researchers only obtained a few results about the Schr ödinger-Poisson system with critical exponent on bounded domain, see for instance [1], [6][7][8][9][10][11].…”

“…In [9], let η = λ, f (x,u) = λ u r , λ > 0 and 0 < r < 1, the author got that system (1.1) has at least two positive solutions and one of the solutions is a ground state solution for all λ ∈ (0,λ * ), where λ * is a positive constant. In [10], assuming that η = −1, f (x,u) = λ u r , λ>0 and 0<r <1, the authors proved that system (1.1) has at least two positive solutions for all λ ∈ (0,λ * ), where λ * is a positive constant. In [11], let η = 1, f (x,u) = λ |x| β u r , λ > 0, 0 < r < 1 and 0 ≤ β < 5+r 2 , combining with the variational method and Nehari manifold method, two positive solutions of system (1.1) are obtained.…”