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Infinitely many solutions for a zero mass Schr\"{o}dinger-Poisson-Slater problem with critical growth
Zhisu Liu,Liu Yang
Keywords:Schr\"{o}dinger-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: $$ -\Delta u+(u^{2}\star \frac{1}{|4\pi x|})u=\mu k(x)|u|^{p-1}u+|u|^{4}u\,\,\mbox{in}\,\,\R^{3}. $$ 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 $\mu^{\ast}>0$.