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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} …

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 the celebrated paper written by Congming Li and Wenxiong Chen, Classification of solutions of some nonlinear elliptic equations, which considered $$\Delta u = -e^u \ \ in \ \ \mathbb{R}^2. …

[This question is looking at the paper Yau, S.-T., On The Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampére Equation, I, Comm. Pure Appl. Math., 31 (1978) 339-411, doi:10.1002 …

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} …

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)}+\ …

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 …

Let $M$ be a bounded domain in $\Bbb R^2$: under the assumption that $$ \partial_{u} f(x, u)=0 \text { for }|u| \geq K\label{1}\tag{1.6} $$ Michael E. Taylor said that (proposition (1.3)) For $k=1,2, …

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 …

I am reading the paper [1] by Congming Li. I want to talk about the typical case that the author gives as follows ([1], §1, pp. 590-): In this section, we study positive solutions of the following se …

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 …

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_ …

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'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 …