Let $\Omega$ be a bounded smooth domain,
$Lu = D\_i \left( a^{ij} (x) D\_ju \right)$, and two constants
$\lambda, \Lambda > 0$. Suppose the coefficient $a$ is measurable,
symmetric, and satisfies
$$
a^{ij} \xi\_i \xi\_j \ge \lambda \vert{\xi} \vert^2 \quad \text{ and} \quad
\sum\_{i,j}^{} \vert{a^{ij}(x)}\vert \le \Lambda^2,
$$
for all $x \in \Omega, \xi \in \mathbb{R}^n$. **However, $a$ can be discontinuous and not belong to any $VMO$ or $BMO$ spaces.**


Let $u \in W^{1,2}(\Omega)$ be a weak solution of $Lu = g$ for
$g \in L^{q/2},~ q > n$. Denote by
$\mathtt{data} = (\lambda,\Lambda,n,q)$. Then Theorem 8.15 in
Gilbarg-Trudinger, *Elliptic Partial Differential Equations of Second
Order*, says that
$$
\Vert u \Vert\_{L^{\infty}(\Omega)} \le C (\mathtt{data}) \left( \Vert u \Vert\_{L^2(\Omega)} + \Vert g \Vert\_{L^{q/2}(\Omega)} \right),
$$
and Theorem 8.24 in the same book says that for any $\Omega' \subset
\Omega$,
$$
\Vert u \Vert\_{C^{\alpha} \left( \Omega' \right)} \le K (\mathtt{data}, \text{dist}(\Omega',\Omega)) \left( \Vert u
\Vert\_{L^2(\Omega)} + \Vert g \Vert\_{L^{q/2}(\Omega)}\right).
$$


On page 214, after Theorem 8.37, the authors claim that solutions $w
\in W^{1,2}\_0(\Omega)$
of the eigenvalue problem
$$
Lw + \sigma w = 0,
$$
belong to $L^{\infty} (\Omega) \cap C^{\alpha}(\Omega)$, thanks to the
above theorems 8.15 and 8.24.


**My question:** If $2 \le n \le 4$, then by
Sobolev embedding, we have $w \in W^{1,2}\_0 \hookrightarrow L^{q/2}$
for some $q > n$. Therefore, we can apply theorems 8.15 and 8.24. What
about the case $n \ge 5$?


Any insights or references are appreciated!
Thank you.



Regularity of eigenfunctions of a self-adjoint differential operator in Gilbarg-Trudinger
I recommend [Luigi Orsina's Lecture Notes](https://www1.mat.uniroma1.it/people/orsina/AS1213/AS1213.pdf). They are beautifully written, and page 24 you will read Stampacchia's approach, which is (in my view) more elegant than Moser iterations and gives you the result you need in a jiffy.


