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

By Zhisu Liu, Zhitao Zhang, Shuibo Huang. Journal of Differential Equations, 2019

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Abstract In this paper we study the following type of the Schrodinger–Poisson–Slater equation with critical growth − △ u + ( u 2 ⋆ 1 | 4 π x | ) u = μ | u | p − 1 u + | u | 4 u , in R 3 , 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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