I have some problems when I read the paper "Symmetry of Positive Solutions of Nonlinear Elliptic Equations in $\mathbb{R}^N$"
The equation (1.3) is $-\Delta u+m^2u=g(u)\quad\text{in} \mathbb{R}^n,\quad n\geq2,m>0$. In the following discusstion, we always assume $m=1$
My problems are from the proof of proposition 4.1
- After $h(r)=O(e^{-ar})$ is obtained, why $$\int_{|x-y|<1}|f(y)|dy=O(e^{-a|x|})$$ it seems that $f(y)=-\Delta u(y)$.
- How to obtain formular (4.6).
From the former formular, we can obtain $$\int_{|x-y|<1}|\Delta u|dy=O(e^{-a|x|})$$. It seems that $$\int_{|x-y|<1} u dy=O(e^{-a|x|})$$ - What is the "standard interior estimates" used after (4.6).
I know that if $Lu=-\sum_{i,j=1}^n(a^{ij}(x)u_{x_i})_{x_j}+\sum_{i=1}^nb^i(x)u_{x_i}+c(x)u$ and $a^{ij}\in C^1(U),b^i,c\in L^\infty(U)$ and $f\in L^2(U)$, the weak solution $u\in H^1(U)$ of $Lu=f$ is $u\in H_{loc}^2(U)$, and for each open $V\subset\subset U$, we have $||u||_{H^2(V)}\leq C(||f||_{L^2}+||u||_{L^2})$.
Is this the "standard interior estimates"? - Why $$\left(\int_{|x-y|<3/4} u^pdy\right)^{\frac{1}{p}}=O(e^{-a|x|}),\quad p=\frac{n}{n-2}$$
It seems that Sobolev embedding is used. But we seem to prove that $||u||_{W^{2,1}(B^0(x,3/4))}=O(e^{-a|x|})$
and whyWhy $$\left(\int_{|x-y|<3/4} |\Delta u|^pdy\right)^{\frac{1}{p}}=O(e^{-a|x|})$$ - How is the "interior estimates" used again to get $$\left(\int_{|x-y|<1/2} u^qdy\right)^{\frac{1}{q}}\subset O(e^{-a|x|}),\quad q=\frac{np}{n-2p}$$ and what does the notation $\subset$ mean.
- How does the formular $u(x)=O(e^{-a|x|})$ obtain.
I would be very grateful if you could help me.
