Multibump solutions for nonlinear Schrödinger-Poisson systems

In this paper, we study the following Schrödinger-Poisson equations: $$\left\{\begin{array}{ll}-{\epsilon }^2\mathrm{\Delta }u+V(x)u+K(x)\varphi u=|u{|}^{p-2}u,& x\in {\mathbb{R}}^3,\\ -{\epsilon }^2\mathrm{\Delta }\varphi =K(x)u^2,& x\in {\mathbb{R}}^3,\end{array}\right.$$ where $p\in (4,6)$, $\epsilon >0$ is a parameter and $V$ and $K$ satisfy the critical frequency conditions. By using variational methods and penalization arguments, we show the existence of multibump solutions for the above system. Furthermore, the heights of these bumps are different order.     

Two solutions for a class of singular Kirchhoff‐type problems with Hardy–Sobolev critical exponent II

In this article, we devote ourselves to investigate the following singular Kirchhoff‐type equation: $$\left\{\begin{array}{l}?\left(a+b{\int }_{\mathrm{\Omega }}|?u{|}^{2}dx\right)\mathrm{\Delta }u=\frac{u^{5?2s}}{|x{|}^s}+\frac{\lambda }{|x{|}^{\beta }{u}^{\gamma }},x\in \mathrm{\Omega },\\ u>0,x\in \mathrm{\Omega },\\ u=0,x\in ?\mathrm{\Omega },\end{array}\right.$$ where $\mathrm{\Omega }?{?}^3$ is a bounded domain with smooth boundary ?Ω,0∈Ω,a≥0,b,λ>0,0<γ,s<1, and $0\le \beta <\frac{5+\gamma }{2}.$ By using the variational and perturbation methods, we obtain the existence of two positive solutions, which generalizes and improves the recent results in the literature.