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


35J60 Nonlinear elliptic equations
35J50 Variational methods for elliptic systems
47J30 Variational methods involving nonlinear operators
Full Text: DOI


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