# Regularity of eigenfunctions of a self-adjoint differential operator in Gilbarg-Trudinger

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.

I'm sorry I ignore the condition of $$a_{ij}$$, but there is a Moser's iteration for $$-\Delta u + V(x)u=0$$. For instance, see Struwe's Variational Methods Appendix B lemma B.3. And I think this iteration doesn't need the continuous of $$a$$.
More precisely, integral the equation $$Lu+cu=0$$ by $$u\min\{|u|^{2s}, M^2\}\in H_{0}^{1}(\Omega)$$, then we have $$\int a^{ij}u_i u_j \min\{|u|^{2s}, M^2\} +\frac{s}{2}\int _{|u|^s\leq M} a^{ij} (|u|^2)_i (|u|^2)_j|u|^{2s-2}\leq c\int |u|^2\min\{|u|^{2s},M^2\}.$$ For $$s$$ small enough such that $$u\in L^{2+2s}(\Omega)$$ we have $$\int a^{ij}u_i u_j \min\{|u|^{2s}, M^2\} \text{ is bounded uniformly with} ~M.$$ Hence let $$M\to \infty$$, we have $$\int |D|u|^{s+1}|^2<\infty,$$ which means, by Sobolev embedding, $$u\in L^{\frac{(2s+2)n}{n-2}}(\Omega)$$. By iteration, you can obtain that $$u\in L^{q}(\Omega)$$ for all $$q>1$$.