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Volume 9, Number 5, 2019, Pages 1706-1718                                                                DOI:10.11948/20180273
Infinitely many solutions for a zero mass Schodinger-Poisson-Slater problem with critical growth
Liu Yang,Zhisu Liu
Keywords:Schrodinger-Poisson-Slater problem, Zero mass, critical growth, concentration-compactness principle.
Abstract:
      In this paper, we are concerned with the following Schr\"{o}dinger-Poisson-Slater problem with critical growth: $$ -\Delta u+(u^{2}\star \frac{1}{|4\pi x|})u=\mu k(x)|u|^{p-2}u+|u|^{4}u\,\,\mbox{in}\,\,\R^{3}. $$ We use a measure representation concentration-compactness principle of Lions to prove that the $(PS)_{c}$ condition holds locally. Via a truncation technique and Krasnoselskii genus theory, we further obtain infinitely many solutions for $\mu\in(0,\mu^{\ast})$ with some $\mu^{\ast}>0$.
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