## 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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### References:

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