Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 4 / b / Solution

On a ball $B_{2r}(x_0)⋐B$, bothare nonnegative solutions and hence supersolutions. Combining the Weak Harnack inequality with the local boundedness estimate for subsolutions gives an oscillation decay estimatewhere $\theta$ depends only on $n,\lambda ,\Lambda$. Iteration yields Hölder continuity with some exponent $\mu \in (0,1)$. The local $L^2$-to-$L^{\infty }$ estimate controls the initial oscillation and gives