## The Schrödinger-Poisson equation under the effect of a nonlinear local term.(English)Zbl 1136.35037

The main object of the paper is the Schrödinger-Poisson problem
$-\Delta u+u+ \lambda\varphi u= u^p, \qquad -\Delta\varphi= u^2, \qquad \lim_{|x|\to\infty}\varphi(x)=0,$
where $$u,\varphi:\mathbb R^3\to\mathbb R$$ are positive radial functions, $$\lambda>0$$ and $$1<p<5$$. Its solutions coincide with critical points of the associated energy functional
$I_\lambda(u)= \int\frac12 \left(|\nabla u(x)|^2+ u^2(x))+ \frac14 \varphi_u(x) u^2(x)- \frac{1}{p+1} |u(x)|^{p+1} \right)\,dx,$ considered for $$u$$ from the Sobolev spaces $$H_r^1$$.
Existence and nonexistence results are contained in the following table:
$\begin{matrix}\l&\qquad\l&\qquad\l&\qquad\l&\qquad\l\\ &\lambda\text{ small} &&\lambda\geq 1/4&{}\\ 1<p<2 &\text{Two solutions} &\inf I>-\infty &\text{No solutions} &\inf I=0 \\ p=2 &\text{One solution} &\inf I=-\infty &\text{No solutions} &\inf I=0 \\ 2<p<5 &\text{One solution} &\inf I=-\infty &\text{One solution} &\inf I=-\infty \end{matrix}$
i.e., $$p=2$$ is the article value for the existence of solutions

### MSC:

 35J60 Nonlinear elliptic equations 35J50 Variational methods for elliptic systems 47J30 Variational methods involving nonlinear operators
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### References:

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