# Critical elliptic equation; kernel of linearized operator

I am interested in the critical equation $- \Delta w(x) = w(x)^p$ in $R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin (and there is an explicit formula). Lets assume $w(x)>0$ is the radial solution of the above problem with the added condition that $w(0)=1$.

Define $L(\phi)= - \Delta \phi - p w(x)^{p-1} \phi$. My interest is in the kernel of $L$ and lets not worry about the exact domain of $L$.

So we expect that $\phi_i:=w_{x_i}$ is in the kernel of $L$ for each $1 \le i \le N$. But also note the original equation is scale invariant and lets assume that $w_\lambda(x)$ satisfies the equation (without the constraint $w_\lambda(0)=1$)) for all $\lambda >0$ with $w_1(0)=1$. Then $\phi_{N+1}:=\partial_\lambda w_\lambda(x)|_{\lambda=1}$ is also in the kernel of $L$.

So my question is: is the kernel of $L$ given by $\{\phi_i : 1 \le i \le N\}$ or $\{ \phi: 1 \le i \le N+1\}$ or neither.

This answer is well known (just not by me and I haven't been able to locate the answer). I attempted to see if $\phi_{N+1}$ is really a linear combination of the other terms (recall there is an explicit formula for $w(x)$ but I am getting bog down in computations).

thanks

NEW QUESTION

Here is the full question I am really interested in. Again I believe this is known material but I can't seem to find the explicit references and my linear analysis is too weak to figure this out. Let $W(x)=w(x)$ be as above and we let $W_{\lambda,z}(x)$ denote $W(x)$ dialated by $\lambda>0$ and translated so its maximum is at $z \in R^N$. Let $V_{\lambda,z}$ dennote the projection of $W_{\lambda,z}$ onto $H_0^1(\Omega)$. So $V_{\lambda,z}(x)$ solves $-\Delta V_{\lambda,z}(x)= W_{\lambda,z}(x)^p$ in $\Omega$ with $V_{\lambda,z} =0$ on \partial \Omega$. \ I am interested in the following linear operator: $$L_{\lambda,z}(\phi)= -\Delta \phi - p V_{\lambda,z}^{p-1} \phi,$$ on certain weighted spaces which i now define. $$\| \phi\|_{X_\lambda}:=\sup_{x \in \Omega} (1 + \lambda^2 |x-z|^2 )^\frac{N-2}{4} \lambda^\frac{-(N-2)}{2} | \phi(x)|$$ $$\| f\|_{X_\lambda}:=\sup_{x \in \Omega} (1 + \lambda^2 |x-z|^2 )^\frac{N+2}{4} \lambda^\frac{-(N+2)}{2} | f(x)|$$ and we let$ X_\lambda$and$Y_\lambda$denote the completion of some suitable spaces under the appropriate norms; the functions in$ X_\lambda$should satisfy a zero Dirichlet boundary condition. This problem is coming up since I am attempting to solve some nonlinear PDE and I need to understand the linear theory on these spaces. In particular I would like a theory which is uniform in$ \lambda$for large$\lambda$. Lets assume$ B_1 \subset \subset \Omega$. So I would like a theory along the lines of: there is some$C>0$and$ \lambda_0>0$such that for all$ |z|<1$,$ \lambda > \lambda_0$and$ f \in L^\infty(\Omega)$there is some$ \phi_{\lambda,z} \in X_\lambda$such that$L_{\lambda,z}(\phi_{\lambda,z})=f$in$ \Omega$and $$\| \phi_{\lambda,z}\|_{X_\lambda} \le C \|f \|_{Y_\lambda}.$$ The important property I would like is the uniform bounds in$ \lambda$and$z$. Now I would assume this is infact not true. Define $$\phi_i := \partial_{z_i} V_{\lambda,z} \quad 1 \le i \le N, \qquad \phi_{N+1}:=\partial_{\lambda} V_{\lambda,z}$$. It appears that the functions$ \phi_i \; 1 \le i \le N+1$should almost be in the kernel of$L_{\lambda,z}$for large$ \lambda$. Lets decompose$X_\lambda$by$ X_\lambda= \tilde{X}_\lambda + span\{ \phi_i: 1 \le i \le N+1\}$and let$\tilde{Y}_\lambda$denote the range of$L_{\lambda,z}$restricted to$\tilde{X}_\lambda $and we should be able to write$ Y_\lambda= \tilde{Y}_\lambda + span\{ \psi_i: 1 \le i \le N+1\}$. QUESTION. Is it plausable that one would have the following linear theory: there is some$ C>0$and$ \lambda_0 >0$such that for all$ f \in L^\infty(\Omega)$,$\lambda > \lambda_0$and$ |z|<1$there is some$ \phi_{\lambda,z} \in \tilde{X}_\lambda$and$ c_i \in R$such that $$L_{\lambda,z}( \phi_{\lambda,z})= f - \sum_{i=1}^{N+1} c_i \psi_i$$ and one has a uniform estimate of the form $$\| \phi_{\lambda,z}\|_{X_\lambda} \le C \|f\|_{Y_\lambda}$$. ($c_i$and$\psi_i$will also depend on$z$and$ \lambda$). I realize there are a lot of details. The main issue I am having trouble with is exactly how to mod out the terms which are almost in the kernel for large$ \lambda$and also how to mod out on the range... Anyhow comments would be greatly appreciated. thanks • It is easy to see that$\phi_{N+1}$cannot be a linear combination of the others. Note that$\phi_{N+1}$is radially symmetric, and all the others have zero average over the sphere. Jul 18, 2013 at 19:33 • Honestly, I'm having trouble following your definition of$\phi_{N+1}$. However, it sounds like you mean the result of the dilation operator acting on the solution$w$. If that's the case, then all of these$\phi_i$should be in the kernel of$L$. This should follow from the general fact that symmetry generators acting on solutions create linearized solutions, which is fairly straight forward to work out for yourself. Jul 18, 2013 at 20:22 • Michael. Thanks, that is a nice way to see it. Jul 18, 2013 at 21:31 • Igor, I am not sure of the correct way of saying this in words...$w_\lambda(x)$is a dilated version of$w(x)$and one can check that$-\Delta w_\lambda(x)=w_\lambda(x)^p$. Also$ w_1(x)=w(x)$and hence taking derivative in$ \lambda$and setting$ \lambda=1$shows that$ \partial_\lambda w_\lambda(x)|_{\lambda=1}$is in the kernel of$L\$. I was confused as to whether it was independent of the other terms. Jul 18, 2013 at 21:36