Let $\Omega$ be a bounded domain in $\mathbb{R}^N$ with boundary of class $C^2$. It is known that there exists a sequence of eigenvalues $0<\lambda_1<\lambda_2\leq\ldots\leq\lambda_j\leq\ldots$ and a sequence of corresponding eigenfunctions $\{\varphi_j\}_{j\in\mathbb{N}^*}$ for Dirichlet problem $$-\Delta u=\lambda u,\quad u|_{\partial\Omega}=0$$
How to show that $\varphi_j\in W^{2,p}(\Omega)$, with some $p>N$?
We know that the solution $u \in H_0^1(\Omega)$ of $$-\Delta u = f$$ satisfies $u \in W^{2,p}(\Omega)$ for $f \in L^p(\Omega)$. Now, you can take $u = \varphi_j$ and $f = \lambda_j \, \varphi_j$.