Existence and concentration of positive ground state solutions for Schrödinger-Poisson systems (Q2866671)

The paper deals with the existence and concentration of positive ground states for the following semilinear Schrödinger-Poisson system NEWLINE\[NEWLINE \begin{cases} -\varepsilon^2\Delta u +u +\lambda \phi(x)u=b(x)f(u) & ,x \in {\mathbb R}^3,\\ -\varepsilon^2 \Delta \phi=u^2 & ,u\in H^1({\mathbb R}^3),\;x\in {\mathbb R}^3, \end{cases} NEWLINE\]NEWLINE where \(\varepsilon>0\) is a small parameter, \(\lambda\neq0\) and \(f\) is a continuous, superlinear and subcritical nonlinearity. Assuming \(b(x)\) attains a maximum, the authors prove that the system has a positive ground state \((u_\varepsilon,\phi_{u_\varepsilon})\) for all \(\lambda\neq0\) and all small enough \(\varepsilon>0.\) It is shown also that, for each \(\lambda\neq0,\) \(u_\varepsilon\) converges to the positive ground state solution of the associated limit problem and concentrates to a maximum point of \(b(x)\) as \(\varepsilon\to0.\) Sufficient conditions for nonexistence of positive ground states are obtained as well.NEWLINENEWLINEReviewer's remark: What is strange is that the paper under review is a particular case of the more general problem NEWLINE\[NEWLINE \begin{cases} -\varepsilon^2\Delta u +a(x)u +\lambda \phi(x)u=b(x)f(u) & x \in {\mathbb R}^3,\\ -\varepsilon^2 \Delta \phi=u^2 & u\in H^1({\mathbb R}^3),\;x\in {\mathbb R}^3, \end{cases} NEWLINE\]NEWLINE studied by the same authors in [Calc. Var. Partial Differ. Equ. 48, No. 1--2, 243--273 (2013; Zbl 1278.35074)].