Standing waves for Schrödinger-Poisson system with general nonlinearity
Zhi Chen, Xianhua Tang, Ning Zhang, Jian Zhang · Discrete and Continuous Dynamical Systems · 2019
In this paper we consider the following Schrödinger-Poisson system with general nonlinearity \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{ll} -\varepsilon^2\Delta u+V(x)u+\psi u = f(u),\,\, x\in\mathbb{R}^3,\\ -\varepsilon^2\Delta\psi = u^2,\,\,u>0,\,\, u\in H^1(\mathbb{R}^3),\\ \end{array} \right. \end{eqnarray*} $\end{document} where $ \varepsilon>0 $ is a small positive parameter. Under a local condition imposed on the potential $ V $ and general conditions on $ f, $ we construct a family of positive semiclassical solutions. Moreover, the concentration phenomena around local minimum of $ V $ and exponential decay of semiclassical solutions are also explored. We do not need the monotonicity of the function $ u\rightarrow\frac{f(u)}{u^3} $, and our results include the case $ f(u) = |u|^{p-2}u $ for $ 3<p<6 $. Since without more global information on the potential, in the proofs we apply variational methods, penalization techniques and some analytical techniques.