On some regularity properties of mixed local and nonlocal elliptic equations

This article is concerned with “up to $C^{2,\alpha }$-regularity results” about a mixed local-nonlocal nonlinear elliptic equation which is driven by the superposition of Laplacian and fractional Laplacian operators.

First of all, an estimate on the $L^{\infty }$ norm of weak solutions is established for more general cases than the ones present in the literature, including here critical nonlinearities.

We then prove the interior $C^{1,\alpha }$-regularity and the $C^{1,\alpha }$-regularity up to the boundary of weak solutions, which extends previous results by the authors (Su et al., 2022, [20]), where the nonlinearities considered were of subcritical type.

In addition, we establish the interior $C^{2,\alpha }$-regularity of solutions for all $s\in (0,1)$ and the $C^{2,\alpha }$-regularity up to the boundary for all $s\in (0,\frac{1}{2})$, with sharp regularity exponents.

For further perusal, we also include a strong maximum principle and some properties about the principal eigenvalue.