Search Results

I'm reading the paper Remarks concerning the conformal deformation of riemannian structures on compact manifolds by NEIL S. TRUDINGER. I'm stuck with the Theorem 3, which says that let $u$ be a $W_{2} …

My questions arise from Here, it seems that I didn't give a clear question, so I rephrase my questions here. For example, for $$ -\Delta u=f(u) \quad \text { in } \Omega, $$ we call a solution is stab …

Why $$ -\Delta_g u+\lambda=\lambda \frac{e^{2 u}}{\int_M e^{2 u} d \mu_g} $$type PDE is called 'mean-field equation'? It's closely related to moser-trudinger inequality, there are many classical pap …

I just read the celebrated paper Farina and Dancer, which talks about the following PDE in $\mathbb{R}^n$　 $$\Delta u + e^u=0.$$ They proved that when $3 \le n \le 9$, there is no finite Morse index s …

For $$ -\Delta u=f(u) \quad \text { in } \Omega, $$ we call a solution is stable if $$ Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall …

I wonder if there are theories on elliptic operator $$Lu=\Delta u + b_iu_i + cu$$ when $c>0$, when $c<0$, we are glad to have maximum principle, so the bijectivity can be easily analyzed, but I hardly …

I'm reading the Theorem2 in UNIFORM ESTIMATES AND BLOW-UP BEHAVIOR FOR SOLUTIONS OF $-\Delta u=V(x) e^u$ IN TWO DIMENSIONS They prove that for the solution of $$ -\Delta u= V(x)\exp u \text { in } \ma …

Recently I got to know about Pohozaev's identity, and I calculated several examples. The basic idea is multiplying $x \cdot \nabla u$ on both sides of the equation, but I noticed that all the materia …

I'm reading Dr.Luca Martinazzi's paper Existence of solutions to a higher dimensional mean-field equation on manifolds which proves that for $m \geq 1$, there is an existence result for the equation $ …

Recently I'm learning the use of moving plane method to prove radial symmetry of $C^{2}$ global solution of a PDE in $R^{2}$, and I'm reading a paper where this method is applied: precisely I'm readin …

In this part of the book "Elliptic PDE" of Qing Han & Fanghua Lin, the Leray–Schauder existence theorem is applied to prove the existence of $C^{2, \alpha}(\bar{\Omega})$ solution. For $\beta \in(0,1) …

Let $\sigma \in[0,1]$,we consider following series of linear partial differential equations related to the parameter $\sigma$,for example $$ \left\{\begin{aligned} \Delta \Phi &=\sigma f(x, y) \text { …

I'm reading Prof. Kazdan's lectures At page 69, Prof. Kazdan describes the research on the $\Delta u=c-h e^{u}$ PDE on a compact $n$-dimensional manifold before 1983. (Here $c$ is a constant while $h$ …

Recently I see a question linear second order PDE in which user Pedro post a reference in Gilbarg's book, which said that the solvability of the linear PDE $$ \Delta u +B^{i}(x)u_{i}+C(x)u=f $$ is equ …

If a Dirichlet problem (elliptic PDE, in $R^{n}$) is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold? For example, $B$ is the unit ball in $R^{n}$, the G …