Shuangjie Peng

This paper deals with the following Cauchy problem for critical heat equation with drift term $ \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u +\nabla \ln a(x)\cdot \nabla u +|u|^{\frac{4}{n-2}}u, & \mbox{for}\ \ &(x, t)\in \mathbb{R}^n\times (0, +\infty), \\ &u(\cdot, 0) = u_0, & \mbox{for}\ \ &x\in \mathbb{R}^n, \end{aligned} \right. \end{equation*} $ where $ a(x) $ is a positive smooth bounded function in $ \mathbb{R}^n, n\geq 6 $. Assume that the eigenvalues of matrix $ A(q) $, denoted as $ \sigma_i, \; i = 1, \cdots, n $, satisfy $ \begin{equation*} \sigma_i<(1+\sigma)\frac{3n}{n+2}\sum\limits_{i = 1}^{n}\frac{\sigma_i}{n}. \end{equation*} $ We construct sign-changing bubble tower solutions which blow up at the critical point $ q $ of $ a(x) $ in the forward time infinity of the form $ u(x, t)\approx \sum\limits_{j = 1}^{2}(-1)^{j-1}\alpha_n \bigg(\frac{\mu_j}{\mu_j^2+|x-q|^2}\bigg)^{\frac{n-2}{2}} \ \ \text{as}\ \ t\rightarrow +\infty, $ here $ 0<\mu_j(t)\rightarrow 0 $ exponentially as $ t\rightarrow +\infty. $

This paper is concerned with normalized solutions of the magnetic focusing Gross-Pitaevskii equations with anharmonic potentials in $\R^N$, where $N=2,3$. The existence of axially symmetric solutions is constructed as the parameter $a>0$ satisfies $a \to a_*(N)$, where $a_*(N)\geq0$ is a critical constant depending only on $N$. We further prove that up to the constant phase and rotational transformation, normalized concentrating solutions as $a\to a_*(N)$ must be unique and axially symmetric. As a byproduct, we also obtain that for the case $N=3$, the normalized concentrating solution as $a\to a_*(3)$ is free of vortices, where the anharmonic potential is non-radially symmetric.

We consider the following supercritical problem for the Lane-Emden system: \begin{equation}\label{eq00} \begin{cases} -Δu_1=|u_2|^{p-1}u_2\ &in\ D,\\ -Δu_2=|u_1|^{q-1}u_1 \ &in\ D,\\ u_1=u_2=0\ &on\ \partial D, \end{cases} \end{equation} where $D$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq4.$ What we mean by supercritical is that the exponent pair $(p,q)\in(1,\infty)\times(1,\infty)$ satisfies $\frac1{p+1}+\frac1{q+1}<\frac{N-2}N$. We prove that for some suitable domains $D\subset\mathbb{R}^N$, there exist positive solutions with layers concentrating along one or several $k$-dimensional sub-manifolds of $\partial D$ as $$\frac1{p+1}+\frac1{q+1} \rightarrow \frac{n-2}{n},\ \ \ \ \frac{n-2}{n}<\frac1{p+1}+\frac1{q+1}<\frac{N-2}N,$$ where $n:=N-k$ with $1\leq k\leq N-3$. By transforming the original problem \eqref{eq00} into a lower $n$-dimensional weighted system, we carry out the reduction framework and apply the blow-up analysis. The properties of the ground states related to the limit problem play a crucial role in this process. The corresponding exponent pair $(p_0,q_0)$, which represents the limit pair of $(p,q)$, lies on the critical hyperbola $\frac n{p_0+1}+\frac n{q_0+1}=n-2$. It is widely recognized that the range of the smaller exponent, say $p_0$, has a profound impact on the solutions, with $p_0=\frac n{n-2}$ being a threshold. It is worth emphasizing that this paper tackles the problem by considering two different ranges of $p_0$, which is contained in $p_0>\frac n{n-2}$ and $p_0<\frac n{n-2}$ respectively. The coupling mechanisms associated with these ranges are completely distinct, necessitating different treatment approaches. This represents the main challenge overcome and the novel element of this study..