相关论文: The Nirenberg problem on high dimensional half sph…
We study a Dirichlet boundary problem related to the fractional Laplacian in a manifold. Its variational formulation arises in the study of magnitude, an invariant of compact metric spaces given by the reciprocal of the ground state energy.…
In this paper I consider the inverse boundary value problem for a quasilinear, anisotropic, elliptic equation of the form $\nabla\cdot(\gamma\nabla u+|\nabla u|^{p-2}\nabla u)=0$, where $\gamma$ is a smooth, matrix valued, function with a…
We consider the following elliptic system with Neumann boundary: \begin{equation} \begin{cases} -\Delta u + \mu u=v^p, &\hbox{in } \Omega, \\-\Delta v + \mu v=u^q, &\hbox{in } \Omega, \\\frac{\partial u}{\partial n} = \frac{\partial…
In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u =…
In this paper, we study the fractional critical Schr\"{o}dinger-Poisson system \[\begin{cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{in}~{\mathbb…
We study critical metrics of higher-order curvature functionals on compact Riemannian $n$-manifolds $(M,g)$. For an integer $k$ with $2 \leq 2k \leq n$, let $R^k$ denote the $k$-th exterior power of the Riemann curvature tensor. We…
While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less in known about critical points of the corresponding energy. Saddle…
The paper concerns with the decay property of solutions to the initial-boundary value problem of the semilinear heat equation $\partial_tu-\Delta u+u^p=0$ in exterior domains $\Omega$ in $\mathbb{R}^N$ ($N\geq 2$). The problem for the…
For dimensions $n\geq8$, we are concerned with the quotient functional of the biharmonic Br\'{e}zis-Nirenberg problem under the Navier boundary condition $$ S(\varepsilon V):=\inf_{0\not\equiv u\in H^2(\Omega)\cap…
We study analytical and computational aspects for Dirichlet problem on the unit ball $B$: $|x|<1$ in $R^n$, modeled on the equation \[ \Delta u +\lambda \left(u^p+u^q \right)=0, \;\; \mbox{in $B$}, \;\; u=0 \s \mbox{on $\partial B$}, \]…
We first establish local $C^2$ estimates of solutions to the $\sigma_2$-curvature equation with nonlinear Neumann boundary condition. Then, under assumption that the mean curvature of a background metric is nonnegative on totally…
In this paper we consider a mixed Dirichlet-Neumann boundary value problem. lem involving Choquard nonlinearity with upper critical exponent in the sense of Hardy- Littlewood Sobolev inequality. We investigate the effect of the geometry of…
In this paper, we are concerned with the existence and concentration phenomena of solutions for the following singularly perturbed fractional Schr\"{o}dinger problem \begin{align*} \varepsilon^{2s}(-\Delta)^su+V(x)u=f(u) \ \ \ \mbox{in} \ \…
We study a fractional conformal curvature flow on the standard unit sphere and prove a perturbation result of the fractional Nirenberg problem with fractional exponent $\sigma \in (1/2,1)$. This extends the result of Chen-Xu (Invent. Math.…
This article studies the Dirichlet problem for a class of degenerate fully nonlinear elliptic equations on Riemannian manifolds with \textit{mean concave} boundary in the sense that the mean curvature of the boundary is…
In this paper a fourth order equation involving critical growth is considered under Navier boundary condition. We give some topological conditions on a given function to ensure the existence of solutions. Our methods involve the study of…
A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the normal to $\partial\Omega $. As shown…
In this paper, we are concerned with the critical Hartree equation \begin{equation*} \begin{cases} -\Delta u=\left(\displaystyle{\displaystyle{\int_{\Omega}}}\frac{u^{2^{*}_{\mu}}(y)}{|x-y|^{\mu}}dy\right)u^{2^{*}_{\mu}-1}+\varepsilon…
Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods…
We consider the Neumann problem $$(P)\qquad - \Delta v + v= v^{q-1} \ \text{in }\ \mathcal{D}, \ v > 0 \ \text{in } \ \mathcal{D},\ \partial_\nu v = 0 \ \text{on } \partial\mathcal{D} ,$$ where $\mathcal{D} $ is an open bounded domain in…