User su - MathOverflow

A 'conjecture' on critical elliptic pde

2013-01-05T20:32:51Z

I conjecture the following.

Let $\Omega=\mathbb{R}^3-\overline{B_1(0)}$. Define $$E_{\Omega}(u)=\frac{1}{2}\int_{\Omega}|\nabla u|^2dx-\frac{1}{6}\int_{\Omega}|u|^6dx.$$ $E_{\mathbb{R}^3}$ is defined similarly: $$E_{\mathbb{R}^3}(u)=\frac{1}{2}\int_{\mathbb{R}^3}|\nabla u|^2dx-\frac{1}{6}\int_{\mathbb{R}^3}|u|^6dx.$$

Consider the exterior problem $$ \Delta u+|u|^4u=0,~~ ,~~~u|_{\partial\Omega}=0$$

It's well-known that if $\Omega=\mathbb{R}^3$, then the problem has a unique radial positive solution given by $$W(x)=\frac{1}{(1+\frac{|x|^2}{3})^{1/2}}.$$

Conjecture: If $\Omega=\mathbb{R}^3-B_1(0)$, then the problem admits a unique nontrivial nonnegative radial solution $u$. Moreover, $E_{\Omega}(u)=E_{\mathbb{R}^3}(W)$.

I appreciate very much if anynone can prove this or can tell me the existed source of answer or give counterexamples.

A critical elliptic PDE

2013-01-04T01:55:54Z

I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, or $\Omega$ is the light cone.

My question is the existence of solution to this problem.

I understand that this problem has been studied extensively, there should be many results. Yet I still fail to find the answer. So I hope someone can give me a summary of this problem. Thanks!

Best constant of Gagliardo-Nirenberg inequality in exterier domain

2012-12-29T05:36:46Z

In $\mathbb{R}^N$, we know that the best constant of Gagliardo-Nirenberg is characterized by the solution $Q$ of $-\Delta u+u=|u|^2u$ with minimal mass. One have $||u||_4^4\leq C||u||_2||\nabla u||_2^3$, where the constant $C$ is depending on $Q$.

Now if our domain is $\Omega=\mathbb{R}^N\setminus B_1(0)$, i.e., the exterior domain outside the unit ball, then we ask the question, what's the corresponding best constant? I.e., find the best $C_1$ s.t. $\|u\|_{4,\Omega}^4 \le $ $C_1$ $\|u\|_{2,\Omega}$

$\|\nabla u\|_{2,\Omega}^3$.

Comment by Su

2013-02-01T20:31:15Z

Craig, thanks for telling me this technique. I also realized that the answer should be straight forward from the known result. Assume $u$ is a solution to the problem (with certain regularity, for example, $C^1$), then extend $u$ by zero to the whole space, then we must have $u=0$ by the characterization of solutions to the cauchy problem $\Delta u+u^5=0$ in $\mathbb{R}^3$.

Comment by Su

2013-01-05T22:33:45Z

Yes, I want zero boundary condition.