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Positive solutions of nonlinear Schrödinger-Poisson systems with radial potentials vanishing at infinity. (English) Zbl 1200.35098

The paper deals with mountain-pass solutions to a system of Schrödinger–Poisson equations of the form \[ \begin{cases} -\Delta u+V(x)u+\phi u=K(x) u^p,\quad x\in \mathbb R^N,\\ -\Delta \phi=u^2, \end{cases} \] where \(N\in\{3,4,5\},\) \(1<p<(N+2)/(N-2),\) \(V,\;K:\;\mathbb R^N\to \mathbb R_+\) are radial and smooth and satisfy \[ {a\over{1+|x|^\alpha}}\leq V(x)\leq A,\quad 0<K(x)\leq {b\over {1+|x|^\beta}} \] with \(\alpha\in (0,2],\) \(a,A,b,\beta>0.\) Define \[ \sigma=\begin{cases} {{N+2}\over {N-2}}-{{4\beta}\over {\alpha(N-2)}} & \text{if}\;0<\beta<\alpha,\\ 1 & \text{otherwise,} \end{cases} \] and \[ \alpha^*={{2(N-1)(N-2)}\over {3N+2}}. \] Assuming \(\alpha<\alpha^*\) and \(p\in (\sigma,2^*-1)\cap [3,2^*-1),\) the author proves that there exists a non-trivial, positive and radial mountain-pass solution \((u,\phi)\in H^1(\mathbb R^N)\times {\mathcal D}^{1,2}(\mathbb R^n)\) of the above system such that \(\lim_{|x|\to\infty}u=0\).

MSC:

35J50 Variational methods for elliptic systems
35J47 Second-order elliptic systems
35J10 Schrödinger operator, Schrödinger equation
35B09 Positive solutions to PDEs
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