I study by myself about Elliptic Problem in specific domain written by P. Grisvard.(Chapter 4) Let's denote $H^{2}=W^{2,2}$.
Let's assume $u$ is a weak solution in $H^{2}$such that $$ \begin{cases} -\triangle u+au=f & \text{in }\Omega,\\ u=0 & \text{on }\partial\Omega, \end{cases} $$ where $f\in L^{2}(\Omega)$.
Moreover, we fix $\Omega=\{(x,y)|x\in\mathbb{R},\ 0<y<h\}$ which is an infinite strip. In the textbook, the following is satisfied : $$ \|u\|_{H^{2}(\Omega)}\leq C\|f\|_{L^{2}(\Omega)}, $$ for some constant $C$ only when $a>0$.
What if $a=0$? In other words, if $-\triangle u=f$ in $\Omega$, is above statement still hold? If not, I'm looking for the counterexamples
The statement does hold for $a=0,$ where it becomes a form of Poisson's equation.
Consider the extension of $u$ to $\overline u:\mathbb R\times (\mathbb R/2h)\to\mathbb C$ that is odd in the $y$-direction, i.e. $\overline u(x,y)=-\overline u(x,-y).$ By a similar argument as in the proof of the Sobolev extension theorem, this extension is still in $H^2,$ so in fact it must satisfy the equation $-\Delta \overline u=\overline f$ where $\overline f$ is the extension of $f$ that is odd in the $y$-direction. This extension is defined on an abelian group so has a Fourier transform. I will drop the overlines from now on - they were just used to justify the Fourier transform.
The Fourier transform $\hat u$ can be thought of as satisfying $$u(x,y)=\int \sum_{k=1}^\infty \hat u(p,k) e^{i p x}\sin(\pi k y/h) dp$$ so that the $x$ derivative becomes multiplication by $ip$, and the $y$ derivative is like multiplication by $\pi k/h$ (and switching $\sin\mapsto \cos\mapsto-\sin$), so $$\|u\|^2_{H_2(\Omega)}\geq C\int \sum_{k=1}^\infty |\hat u(p,k)|^2 (1+p^2+( k/h)^2+p^4+p^2( k/h)^2+( k/h)^4) dp$$ and $$\|f\|^2_{L_2(\Omega)}\leq C\int \sum_{k=1}^\infty |\hat u(p,k)|^2 (p^4+(k/h)^4) dp$$
where the $C$'s may be different - I am using $C$ to ignore constant factors, and with the right constants these would be equalities. These easily give $\|u\|_{H_2(\Omega)}\leq C(1+h^2) \|f\|_{L_2(\Omega)}$ for some $C$. (Use $1\leq h^2(k/h)^2$ and some "$2ab\leq a^2+b^2$" inequalities.)