Multiple Positive Solutions for the Nonlinear Schrödinger–poisson System
نویسندگان
چکیده
has been studied extensively by many researchers, where 2 < p < 6. This system has been first introduced in [5] as a physical model describing a charged wave interacting with its own electrostatic field in quantum mechanic. The unknowns u and Φ represent the wave functions associated to the particle and electric potential, and functions V and K are respectively an external potential and nonnegative density charge. We refer to [5] and the references therein for more physical background. In recent years, there has been increasing interest in studying problem (1.1). In the case of V (x) ≡ 1, K(x) ≡ λ > 0, the existence of radially symmetric positive solutions of system (1.1) was obtained by D’Aprile and Mugnai in [9] and Ruiz in [20] for p ∈ (3, 6). Azzollini and Pomponio in [4] established the existence of ground state solutions for p ∈ (3, 6). For p ∈ (2, 3), λ = 1, Ruiz in [20] proved that (1.1) does not admit any nontrivial solution. When K(x) ≡ 1 and V (x) is not a constant, the authors proved that there exist radially symmetric solutions concentrate on the spheres in [12, 14] and a positive bound state solution concentrates on the local minimum of the potential V in [13]. By using constrained minimization on the signchanging Nehari manifold and the Brouwer degree theory, Wang and Zhou in [23]
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