What happens to the De Giorgi-Nash-Moser estimate when the potential term lies in the critical $L^{\frac n2}$ space?

Question

I am looking for references/answers that could provide guidance to the following question: Where are some counter-examples to DeGNM for the critical case $q=n/2$ for one of the coefficients of the elliptic PDE, if they exist? That is, if we consider the weak form of the PDE $$ -\text{div }A\nabla u+Vu=f $$ on a ball of radius $1$, are there counter-examples to Moser estimate when $V\in L^{n/2}(B_1)$? Do note that Theorem 4.4 in the book "Elliptic Partial Differential Equations" by Han and Lin gives that $u$ lies in $L^p(B_{1/2})$ for every $p\in[2,+\infty)$, and gives a corresponding Moser-type estimate, but the constant depends on $p$ here, and it blows up as $p\rightarrow\infty$, so it is not clear that the boundedness estimate I am looking for is achievable.

Any related answer is welcome.

Answer 1

Consider $u_{\epsilon} = \rho_{\epsilon} \ast \log$ in $B_{1/2} \subset \mathbb{R}^2$, where $\rho_{\epsilon}$ are standard mollifiers. Then $|u_{\epsilon}(0)|$ blows up as $\epsilon \rightarrow 0$, and $\Delta u_{\epsilon} = \rho_{\epsilon}$ (representation formula) has unit mass independent of $\epsilon$. Taking $V = \rho_{\epsilon}/u_{\epsilon}$ and $f = 0$, we see that the estimate doesn't hold.

Unbounded examples include $u = \log\log(r^{-1})$ in $B_{1/8} \subset \mathbb{R}^n$ for $n \geq 2$ (taking $V = \Delta u / u$ and $f = 0$). In the case $n = 2$ the Laplace is $(r|\log r|)^{-2}$ which is integrable near the origin, and in the case $n \geq 3$ the Laplace is order $r^{-2}|\log r|^{-1}$ which is $L^{n/2}$.

The underlying reason the estimate fails is that $\int |\Delta u|^{n/2}\,dx$ is invariant under the scaling $u \rightarrow u(Rx)$. This gives another approach to constructing examples: for $n = 1$ start with the building block $u_0$ that smoothly connects $0$ to the left and $x$ to the right, and take a weighted sum of translations, e.g. $u = 1 + \sum_{k > 0} k^{-3}u_0(k^3(x + 1/k))$ (a modification of $\log$). (In fact, by summing Lipschitz rescalings $a_k^{-1} u_0(a_k(x + 1/k))$ with $a_k$ as large as we like we can make $\int |u''|^{1/2}$ arbitrarily small).