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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 considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2} \exp u(x) \, d x< +\infty$. LEMMA 1.1 (Ding …

Actually I'm reading a paper on mean-field equation on torus by M.Struwe and G.Tarantello Here, they studied $$\tag{1} -\Delta u=\lambda\left(\frac{e^u}{\int_{\Omega} e^u d x}-\frac{1}{|\Omega|}\right …

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 wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying $$\Delta_c u = f(x,u), …

I'm considering the elliptic PDE with complex Laplacian, for example, write $$ \Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot), $$ and $$\Delta_c(u)=f,$$ by [P.Gauduchon, Math.Ann …

I'm reading a paper which said that the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\ …

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 …

Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the k …

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 …

I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 m} …

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 …

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 …

My question arises from Here. I have a series of eigenvalue equations in $B_R$. $$ -\Delta \phi_R+H(x) \phi_R=\lambda_R \phi_R, $$ where $\lambda_R \geq 0$ is the first nonzero eigenvalue, with $\phi_ …

I want to talk about generalizing the conception of 'stable' solution and 'stable outside the compact set' solution of elliptic PDE from $\mathbb{R}^n$ to closed manifold. I'm reading $\Delta u +e^u=0 …