arXiv subjects
Publications and source records attributed to Pingping Yang.
In this paper, we construct new sign-changing solutions for the H\'{e}non problem \begin{equation*} \begin{cases} -\Delta u= |x|^{\alpha} |u|^{p-1}u,\,\,\,\text{in}\,\,\, \Omega, \\[1mm] u=0,\,\,\,\,\,\,\text{on}\,\,\, \partial\Omega, \end{cases} \end{equation*} where $\Omega\subseteq \mathbb{R}^2$ is a bounded smooth domain containing 0, $\alpha\in\mathbb{R}$ is a positive parameter and $p$ approaches $+\infty$.
In this paper, we consider the following fractional Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{lcl} (-\Delta)^{s}u+V(x)u=u^{{p_s}-\epsilon}\ \ \ &\hbox{in}\ \mathbb{R}^N,\\ u>0\ \ \ &\hbox{in}\ \mathbb{R}^N, \end{array} \right. \end{equation*} where $0 0$, $p_s=(N+2s)/(N-2s)$, $N>4s$ and $V(x)\in C^1(\mathbb{R}^N)\cap L^\infty (\mathbb{R}^N)$ is non-negative. We first use the Lyapunov-Schmidt reduction method to construct multi-peak solutions to the above equation provided that $V(x)$ possesses $k$ stable critical points. Then we prove the non-degeneracy and local uniqueness of the multi-peak solutions, for $\frac{1}{2}<s<1$, $N\geq 6s$, via the blow-up argument based on various local Pohozaev identities. Due to the nonlocal property of the fractional Laplacian, we need to make delicate analysis of the approximate solutions and establish the local Pohozaev identities for the corresponding harmonic extension instead of $u$. This approach not only requires to develop refined estimates for several integrals in the local Pohozaev identities, but also to apply Pohozaev identities through a markedly different way.
In this paper, we are concerned with qualitative properties of multi-peak solutions of the following nonlinear Schr\"{o}dinger equations \begin{equation*} -\Delta u+V(x)u= u^{p-\varepsilon},\,\,\,u>0,\,\,\,\text{in}\,\,\,\mathbb{R}^N, \end{equation*} where $V(x)$ is a nonnegative continuous function, $\varepsilon>0$, $p=\frac{N+2}{N-2}$, $N\geq6$. The existence of multi-peak solutions has been obtained by Cao et al. (Calc. Var. Partial Differential Equations, 64: 139, 2025). The main objective in this paper is to establish the local uniqueness and Morse index of the multi-peak solutions in \cite{CLl1} provided that $V(x)$ possesses $k$ non-degenerate critical points by using the blow-up analysis based on Pohozaev identities.
The nonlinear Schr\"{o}dinger-Newton system \begin{equation*} \begin{cases} \Delta u- V(|x|)u + \Psi u=0, &~x\in\mathbb{R}^3,\\ \Delta \Psi+\frac12 u^2=0, &~x\in\mathbb{R}^3, \end{cases} \end{equation*} is a nonlinear system obtained by coupling the linear Schr\"{o}dinger equation of quantum mechanics with the gravitation law of Newtonian mechanics. Wei and Yan in (Calc. Var. Partial Differential Equations 37 (2010),423--439) proved that the Schr\"{o}dinger equation has infinitely many positive solutions in $\mathbb{R}^N$ and these solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are radially symmetric in the other variables. Duan et al. in (arXiv:2006.16125v1) extended the results got by Wei and Yan and these solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are even in $y_{2}$with one more more parameter in the expression of the solutions.Hu et al. Under the appropriate assumption on the potential function V, Hu et al. in (arXiv: 2106.04288v1) constructed infinitely many non-radial positive solutions for the Schr\"{o}dinger-Newton system and these positive solutions have polygonal symmetry in the $(y_{1}, y_{2})$ plane and they are even in $y_{2}$ and $y_{3}$. Assuming that $V(r)$ has the following character \begin{equation*} V(r)=V_{1}+\frac{b}{r^q}+O\Big(\frac{1}{r^{q+\sigma}}\Big),~\mbox{ as } r\rightarrow\infty, \end{equation*} Where $\frac12\leq q<1$ and $b, V_{1}, \sigma$ are some positive constants, $V(y)\geq V_1>0$, we construct infinitely many non-radial positive solutions which have polygonal symmetry in the $(y_{1}, y_{2})$ plane and are even in $y_{2}$ for the Schr\"{o}dinger-Newton system by the Lyapunov-Schmidt reduction method. We extend the results got by Duan et al. in (arXiv:2006.16125v1) to the nonlinear Schr\"{o}dinger-Newton system.
There is growing evidence to suggest that the black hole mass has been previously underestimated with the H$β$ line width for certain active galactic nuclei (AGN). With the assumption of the flatter rather than isotropic velocity distribution of gases in the broad-line region of AGN, we investigated the properties of narrow line Seyfert 1 (NLS1) galaxies, like the black hole mass and the Eddington ratio, and compared with broad line Seyfert 1 (BLS1) galaxies. Since gamma-rays detected in a few NLS1s which favor a smaller viewing angle in NLS1s than BLS1s, with the projection effect we estimated the relative black hole mass and Eddington ratio for NLS1s and BLS1s. The result implies that the NLS1s and BLS1s have similar black hole masses and Eddington ratios, peaked at a larger black hole mass and lower Eddington ratio for the NLS1s than thought before. Furthermore, with applying the correction factor 6 of average black hole mass as derived from the modelling of both optical and UV data in radio-loud NLS1s by Calderone et al., to the Xu et al. sample, we find that the NLS1s and BLS1s also show similar black hole masses and Eddington ratios, peaked at $2.0\times10^{7}M_{\odot}$ and 0.12 (Eddington ratio) for the NLS1s. The $M_{BH}-σ$ relation due to the enhanced black hole masses of NLS1s is discussed. In addition, there seems to show a linear correlation between jet power and disk luminosity for the flat spectrum radio-loud NLS1 sample, which implies an accretion dominated rather than black hole spin dominated jet.