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Consider the elliptic equation $-\Delta u=\alpha f(u)$ in $R^n$ and assume that for some $\alpha^* \in R$ it has a bounded smooth solution $u^*$ in $R^n$ ($f$ is a nice smooth function). Under what co …

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. Suppose the …

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and assume $u$ is a solutions of $\nabla \cdot (a \nabla u)=f$ with $a>c>0$ in $\Omega$, where $a\in C^k(\bar{\Omega})$. Is the following regularity …

I wonder if the ODE $y''+e^{y}=a$ can be solved explicitly. For $a=0$, it is well-known that there is a two-parameter family of explicit solutions $y=\ln(2)-2\ln(\cosh(cx+d))+2\ln(c)$, $c,d \in R …

Let $a_0$ and $b_0$ be smooth compactly supported functions in $B \subset R^3$, $f\in C^1(\Omega)$, and define $a_n=f\Delta^{-1}(a_{n-1})=-f(x)\int_{B}a_{n-1}(y)\Phi(x-y)dy$, $n\geq 1$ $b_n=f\Delta^ …

Let $\Omega$ be a bounded smooth region in $R^n$ and $u$ satisfy $-\Delta u+a(x)u=f, \ \ u|_{\partial \Omega}=0$, where $a(x)\geq 0$ and $f(x)$ are smooth functions. I wonder if the following estim …

Suppose $\varphi$ is a radial (and radially decreasing) solution of $$\Delta \varphi+e^{\varphi}=0, \ \ \text{on} \ \ r \in (0,R), $$ with $ R>0$, and $\psi$ is a decreasing radial function satisfying …

Let $\alpha \in \mathbb{R}$ and $u_\alpha$ satisfy $$ \Delta u_\alpha+e^{u_\alpha}=\alpha f(x), \ \ \ \ x\in \mathbb{R}^2$$ where $f$ is a fast decaying smooth function. I would like to know how the …

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\ …

Let $f$ be a smooth function on $R^2$, and define $N_f$ to be the set of points $p$ such that the nodal set of $f$ ($\{x\in R^2: f(x)=0\}$) divided every neighborhood of $p$ into four regions. Indeed, …

Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and $\Delta u=f$ in $\Omega$ such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| …

Let $I[u]$ be a functional on a (possibly infinite dimensional) Hilbert space. Then, under some conditions, the Mountain Pass theorem guarantees the existence of a saddle point (see http://en.wikipedi …