Existence and nonexistence of positive solutions for a static Schrödinger–Poisson–Slater equation

Abstract

In this paper we study the following type of the Schrödinger–Poisson–Slater equation with critical growth−△u+(u2⋆1|4πx|)u=μ|u|p−1u+|u|4u,inR3, where μ>0 and p∈(11/7,5). For the case of p∈(2,5). We develop a novel perturbation approach, together with the well-known Mountion–Pass theorem, to prove the existence of positive ground states. For the case of p=2, we obtain the nonexistence of nontrivial solutions by restricting the range of μ and also study the existence of positive solutions by the constrained minimization method. For the case of p∈(11/7,2), we use a truncation technique developed by Brezis and Oswald [9] together with a measure representation concentration-compactness principle due to Lions [27] to prove the existence of radial symmetrical positive solutions for μ∈(0,μ⁎) with some μ⁎>0. The above results nontrivially extend some theorems on the subcritical case obtained by Ianni and Ruiz [18] to the critical case.