Here are three nontrivial examples with uniformly elliptic equations.

**1** Nadirashvilli and Valduts example of a solution to a fully nonlinear uniformly elliptic PDE with constant coefficients which is not $C^2$.

**Nadirashvili, N. and Vlăduţ, S.**
*Nonclassical solutions of fully nonlinear elliptic equations.*
Geom. Funct. Anal. 17 (2007), no. 4, 1283–1296.

The original example was in dimension 12. In a later work (joint with Vladimir Tkachev) they brought the dimension down to 5. The example is known to be impossible in dimension 2. Dimensions 3 and 4 are still open.

**2** Safonov's example of a uniformly elliptic equation in 3D whose solution cannot be Lipschitz continuous.

**Safonov, M. V.**
*Unimprovability of estimates of Hölder constants for solutions of linear elliptic equations with measurable coefficients.* (Russian) Mat. Sb. (N.S.) 132(174) (1987), no. 2, 275--288; translation in Math. USSR-Sb. 60 (1988), no. 1, 269–281

Such example is known to be impossible in dimension 2.

**3.** Plis' example of a uniformly elliptic equation with Holder coefficients for which the unique continuation property does not hold.

**Pliś, A.**
*On non-uniqueness in Cauchy problem for an elliptic second order differential equation.*
Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 11 1963 95–100.
35.42

In particular, he proves that there exists a non zero solution to some uniformly elliptic PDE in 3D with Holder coefficients which is identically zero outside of a ball.

It is related to the example that Hendrik Vogt suggested. Again, in dimension 2 it would not be possible.