Dynamics of Bubbling Wave Maps with Prescribed Radiation

Compactness issues and bubbling phenomena for the prescribed Gaussian curvature equation on the Torus

In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics $g_λ$ of prescribed Gauss curvature $K_{g_λ} = f_0 + λ$ on the torus, when the parameter $λ$ tends to one of the boundary points of the interval of existence of $g_λ$, and we characterize their "bubbling behaviour".

Infinitely many bubbling solutions and non-degeneracy results to fractional prescribed curvature problems

We consider the following fractional prescribed curvature problem $$(-Δ)^s u= K(y)u^{2^*_s-1},\ \ u>0,\ \ y\in \mathbb{R}^N,\qquad (0.1)$$ where $s\in(0,\frac{1}{2})$ for $N=3$, $s\in(0,1)$ for $N\geqslant4$ and $2^*_s=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent, $K(y)$ has a local maximum point in $r\in(r_0-δ,r_0+δ)$. First, for any sufficient large $k$, we construct a $2k$ bubbling solution to (0.1) of some new type, which concentrate on an upper and lower surfaces of an oblate cylinder through the Lyapunov-Schmidt reduction method. Furthermore, a non-degeneracy result of th…

Bubbling of the prescribed $Q$-curvature equation on $4$-manifolds in the null case

Analog to the classical result of Kazdan-Warner for the existence of solutions to the prescribed Gaussian curvature equation on compact 2-manifolds without boundary, it is widely known that if $(M,g_0)$ is a closed 4-manifold with zero $Q$-curvature and if $f$ is any non-constant, smooth, sign-changing function with $\int_M f dμ_{{\it g}_0} <0$, then there exists at least one solution $u$ to the prescribed $Q$-curvature equation \[ \mathbf{P}_{g_0} u = f e^{4u}, \] where $\mathbf{P}_{g_0}$ is the Paneitz operator which is positive with kernel consisting of constant functions. In this paper, we…

Leray-Schauder degree for the resonant Q-curvature problem in even dimensions

In this paper, using the theory of critical points at infinity of Bahri, we derive an exact bubbling rate formula for the resonant prescribed Q-curvature equation on closed even-dimensional Riemannian manifolds. Using this, we derive new existence results for the resonant prescribed Q-curvature problem under a positive mass type assumption. Moreover, we derive a compactness theorem for conformal metrics with prescribed Q-curvature under a non-degeneracy assumption. Furthermore, combining the bubbling rate formula with the construction of some blowing-up solutions, we compute the Leray-Schauder…

Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type

This paper deals with the following critical elliptic systems of Hamiltonian type, which are variants of the critical Lane-Emden systems and analogous to the prescribed curvature problem: \begin{equation*} \begin{cases} -Δu_1=K_1(y)u_2^{p},\ y\in \mathbb{R}^N,\\ -Δu_2=K_2(y)u_1^{q}, \ y\in \mathbb{R}^N,\\ u_1,u_2>0, \end{cases} \end{equation*} where $N\geq 5, p,q\in(1,\infty)$ with $\frac1{p+1}+\frac1{q+1}=\frac{N-2}N$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. At first, under suitable conditions on $K_1,K_2$ and the certain range of the exponents $p,q$, we construct an unbounded s…

具次线性竞争耦合的Brezis--Nirenberg型临界薛定谔系统的分块气泡解

Segregated Bubbling Solutions for a Critical Schrödinger System of Brezis--Nirenberg Type with Sublinear Competitive Coupling

针对N≥5的两分量Brezis--Nirenberg型临界薛定谔系统，通过变分法与Brouwer度理论，构造了具次线性竞争耦合的分块气泡解，克服了相互作用势梯度不可微的障碍。