# linear operator associated with semilinear elliptic pde

I am reading a paper where at some point they analyse the following linear operator:

$$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$

where $C_\lambda(x)>0$ (smooth) in $\Omega$ a bounded domain in $\mathbb{R}^N$ containing the origin and where the function $C_\lambda$ concentrates at the origin as $\lambda$ gets big. They analyse the operator in some weighted $L^\infty$ spaces, where the weights depends on the parameter $\lambda$. Lets denote the spaces by $X_\lambda$ and $Y_\lambda$. For each fixed $\lambda$ the weights are bounded above by one and bounded away from zero by a constant which depends on $\lambda$. As $\lambda$ gets big both the weights concerge to zero at the origin. Here is their result: there is some explicit function $v_\lambda$ which is smooth and zero on the boundary of $\Omega$, and there is some $C>0$ and $\lambda_0 >0$ such that for all $\lambda > \lambda_0$ and all $f \in L^\infty$ there is some $\phi_\lambda$ such that

$$L_\lambda(\phi_\lambda)=f \text{ in } \Omega \text{ with } \phi_\lambda=0 \text{ on } \partial \Omega.$$

Moreover $\phi_\lambda$ is perpendicular to $v_\lambda$ in $H_0^1$. Moreover one has the following estimate:

$$\| \phi_\lambda\|_{X_\lambda} \le C \| f\|_{Y_\lambda}$$

So here is my question. Lets fix $\lambda$ and analyse the operator. Since $L_\lambda$ is self adjoint there should be a basis of orthogonal eigenfunctions for $L^2$ or $H_0^1$. If one of the eigenvalues was zero then there would be some $f \in L^\infty$ such that $L_\lambda(\phi)=f$ would not be solvable. Hence all the eigenvalues must be non zero. But then there is no way one can hope to find a solution $\phi$ for each $f$ which is perpendicular to $v_\lambda$.

I am missing something? Any comments would be greatly appreciated.

thanks Craig

• $\pOm$ should really be $\partial \Omega$. – Craig Jun 18 '13 at 4:28
• I do not see it an the moment (a reference to the paper would be a help), but your argument works only for $f\in L^2$. It is a big restriction to have $f\in L^\infty$. – András Bátkai Jul 1 '13 at 10:12
• Andras. If you are interested I asked a very related question with regards to this material and would greatly appreciate your comments. thanks – Craig Jul 19 '13 at 3:26