Positive solution for a nonlinear stationary Schrödinger-Poisson system in \(\mathbb R^3\). (English) Zbl 1133.35427
Summary: We are concerned with the stationary Schrödinger-Poisson system
\[ \begin{cases} -\Delta u+V(x)u+\lambda\varphi(x)u= f(x,u), \quad x\in\mathbb R^3,\\ -\Delta\varphi= u^2, \quad \lim\limits_{|x|\to+\infty} \varphi(x)=0, \end{cases} \tag{P} \]
where \(\lambda>0\) is a parameter, the potential \(V(x)\) may not be radially symmetric, and \(f(x,s)\) is asymptotically linear with respect to \(s\) at infinity. Under some simple assumptions on \(V\) and \(f\), we prove that the problem (P) has a positive solution for \(\lambda\) small and has no any nontrivial solution for \(\lambda\) large.
\[ \begin{cases} -\Delta u+V(x)u+\lambda\varphi(x)u= f(x,u), \quad x\in\mathbb R^3,\\ -\Delta\varphi= u^2, \quad \lim\limits_{|x|\to+\infty} \varphi(x)=0, \end{cases} \tag{P} \]
where \(\lambda>0\) is a parameter, the potential \(V(x)\) may not be radially symmetric, and \(f(x,s)\) is asymptotically linear with respect to \(s\) at infinity. Under some simple assumptions on \(V\) and \(f\), we prove that the problem (P) has a positive solution for \(\lambda\) small and has no any nontrivial solution for \(\lambda\) large.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35J20 | Variational methods for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
