Regularity of solutions of one kind of elliptic PDE

Question

Consider $\Omega$ a bounded domain with $C^{1}$ boundary and the elliptic PDE $$ -div (F(\nabla u)) = \lambda|u|^{p-2}u,\ u \in W^{1,p}_0 (\Omega). $$ What kind of conditions on $F$ one must have in order to the weak solution of the PDE be in $C^{1,\alpha}$ (the space of $C^{1}$ functions with derivatives being $\alpha$-Hölder continuous)?

Answer 1

In general you would want:

1. Smoothness of $F$, say $C^1$,

2. A growth-condition from above, typically a $p$-growth condition.

3. A quantitative ellipticity condition, that matches the growth condition.

Note that one can also consider growth conditions involving $q \neq p$ (notably see my comment for $q=2$ at the end), but since you are talking about solutions in $W^{1,p}_0(\Omega)$ I will stick to this special case. A model case would be the $p$-Laplacian, where $F(z) = \lvert z \rvert^{p-2} z$.

In general one may assume $F \in C^1(\mathbb R^n)$ (possibly except at the origin) and satisfies the $p$-ellipticity condition $$ (\mu^2 + \lvert z \rvert^2)^{\frac{p-2}2} \lvert \xi \rvert^2 \lesssim F'(z)[\xi,\xi] \lesssim (1 + \lvert z \rvert^2)^{\frac{p-2}2} \lvert \xi \rvert^2 $$ for all $z \in \mathbb R^n \setminus \{0\}$ and $\xi \in \mathbb R^n$. Here $\mu \geq 0$ is a constant, and we view $F'(z)$ as a bilinear form on $\mathbb R^n$.

The case $\mu = 0$ corresponds to a degenerate case, which I've included to also capture the $p$-Laplacian. If $\mu > 0$ the regularity theory is simpler, but in both cases the above assumptions should suffice to obtain $C^{1,\alpha}$ regularity of solutions for some $\alpha \in (0,1)$ (probably with more assumptions when $\mu=0$).

---

The case $\mu>0$ is the classical De Giorgi-Nash-Moser theory. However this is more of a set of techniques rather than an all-encompassing theorem, so you may struggle to find a reference that covers the exact case you may be interested in. A nice reference is Chapters 7 and 8 of:

Giusti, Enrico, Direct methods in the calculus of variations, Singapore: World Scientific. vii, 403 p. (2003). ZBL1028.49001.

While this text focuses on minima of variational integrals, the techniques do apply to elliptic problems as briefly discussed in Section 8.9. The key argument can be found in Section 8.2, and relies on previous sections as follows:

• Show a suitable function $w$ of $\nabla u$ satisfies satisfies a higher differentiability result from Section 8.2.

• Show this $w$ is a "sub-quasi-minima" of the Dirichlet energy, so the local boundedness result of Sections 7.1 + 7.2 apply.

• Use this to deduce refined minimising properties (as $u$ itself is now Hölder continuous), and apply the results of Section 7.3.

Note that aforementioned result does not consider the additional $\lambda \lvert u \rvert^{p-2} u$ term on the right-hand side, which must then be treated via perturbation. This is the content of Sections 8.5-8.8, however it is lengthy as the author considers a much more general setting than what you write.

Note however that we are assuming $\mu > 0$ (excluding the $p$-Laplacian), and the text largely focuses on the case $p \geq 2$. For further results I would suggest looking at the further references cited in the book, or the monograph:

Ladyzhenskaya, O. A.; Ural’tseva, N. N., Linear and quasilinear elliptic equations, Mathematics in Science and Engineering. 46. New York-London: Academic Press. XVIII, 495 p. (1968). ZBL0164.13002.

---

Side-note: in Jack T's answer, a uniform ellipticity condition associated to quadratic growth is stated. In this case one can also use similar techniques (which simplify slightly), however one needs to assume that $p < \frac{2n}{n-2}$ to treat the right-hand side as a perturbative term.

Answer 2

Suppose for the moment that $u \in C^2(\Omega)$ and $F=F(p)\in C^1(\mathbb R^n;\mathbb R^n)$. Then $$ \operatorname{div} (F(\nabla u(x)))= \sum_{i,j=1}^n \partial_{p_j}F_i(\nabla u(x)) \partial_{ij}u(x).$$Hence, if you want to apply elliptic regularity theory, a reasonable assumption would be $F\in C^1(\mathbb R^n;\mathbb R^n)$ (or possibly $F\in \operatorname{Lip}(\mathbb R^n;\mathbb R^n)$) and \begin{align*} \lambda \vert \xi \vert^2 \leqslant \sum_{i,j=1}^n \partial_{p_j}F_i(p)\xi_i \xi_j \leqslant \Lambda \vert \xi \vert^2 \text{ for all }(\xi,p) \in \mathbb R^n \times \mathbb R^n \end{align*} for some $0<\lambda \leqslant \Lambda<+\infty$.