# Regularity of weak solutions to semi-linear elliptic PDEs

Suppose that $$f:\Bbb R^2\to\Bbb R$$ is a continuous non-linearity and consider the following semi-linear elliptic PDE given by: $$-\Delta u=f(x,u),\;\;x\in\Omega\subset\Bbb R^n,\tag{1}\label{1}$$ To avoid the mention of critical Sobolev exponents and to narrow down the scope of the answer, let us assume that $$n=2$$ and $$\Omega$$ is bounded. Assume that $$u\in H^1_0(\Omega)$$ is a weak solution of \eqref{1}.

Q1. Under what assumptions on $$f$$ can we show that (i) $$u\in H^2(\Omega)$$, (ii) recover a classical solution of \eqref{1}?

Remark 1. There are numerous conditions on $$f$$ that guarantee the existence of $$u$$. For $$n=2$$, see this question. For $$n\geq 3$$, see the the book Semi-linear elliptic equations for beginners by Badiale and Serra.

Remark 2. In Evan's Partial Differential Equtions book, Chpater 6 exercise 4, if $$f(x,u)=g(x)+h(u)$$, where $$g\in L^2(\mathbb{R}^n)$$, $$h(0)=0$$, $$h$$ is smooth, and $$h'\geq 0$$, then a solution $$u\in H^1(\Bbb R^n)$$ to \eqref{1} is in fact in $$H^2(\Bbb R^n)$$.

Remark 3. From this paper, where $$\Omega$$ is the unit ball in $$\Bbb R^n$$ with $$n\geq 2$$, and $$f$$ is Lipschitz with $$\partial_uf\geq -M$$ for some $$M\geq 0$$, I quote the following sentence in the introduction:

We remark that solutions to \eqref{1} are already in $$W^{2,p}(\Omega)\cap C^{1,\alpha}(\Omega)$$ for $$p<\infty$$ and $$\alpha<1$$ (since $$f$$ is bounded).

I think I can workout a proof as to why $$u\in H^2(\Omega)$$, but not $$u\in C^{1,\alpha}(\Omega)$$ for $$\alpha<1$$, which brings us to the second question.

Question 2. How does one obtain $$u\in C^{1,\alpha}(\Omega)$$?

Remark 4. In the answer, assume any needed regularity on $$\Omega$$ to obtain regularity of $$u$$.

• If you know that the right hand side $f(x,u)$ is in $L^2$, then $u \in H^2$. Then boundedness of $f$ suffices, as well as $|f(x,u)| \leq a+b|u|$. A bootstrap argument (under similar hypotheses) yields $u \in W^{2,p}$ and then $u \in C^{1,\alpha}$ by Sobolev embedding. Sep 22, 2021 at 7:35

There is a "standard" bootstrap argument which can be used to show regularity for semilinear equations. I sketch it here under the assumption that $$|f(x, u)| \leq C(1 + |u|^p)$$ for some $$0 < p < \frac{2n}{n-2}$$ in $$n \geq 2$$ (I know the question was posed in $$n = 2$$ but it is useful to see the role played by the critical exponent).

Assume we know that $$u$$ is in $$L^q$$ for some $$p < q < \infty$$, and that $$\Omega$$ is of class $$C^{1,1}$$. Then $$\Delta u \in L^{q/p}$$, and so by the Calderon-Zygmund theorem $$u \in W^{2, q/p}$$. The Calderon-Zygmund theorem is not always stated in this form (i.e. on domains), but this version may be found in Gilbarg-Trudinger (Section 9.6). Applying the Sobolev embedding gives that $$u \in L^{q'}$$, where $$q' = \frac{nq}{np - 2q}$$ (or $$\infty$$ if $$2q > np$$).

Notice that $$q' > q$$ if and only if $$q > \frac{n}{2}(p - 1)$$, and moreover $$q'/q$$ is an increasing function of $$q$$. This means that as long as we start with $$u \in L^q$$ for $$q > \frac{n}{2}(p - 1)$$, we may apply this procedure repeatedly to get that $$u \in L^{q_*}$$ for any $$q_* < \infty$$, with the number of steps needed always finite and depending only on $$q, q_*$$. For a sufficiently large value of $$q_*$$, $$u \in W^{2, q_*}$$ implies $$\nabla u \in C^{0, \alpha}$$ for $$\alpha \in (0, 1)$$, from Sobolev embeddings.

The point, then, is to check that $$u \in L^q$$ for $$q > \frac{n}{2}(p - 1)$$. This is not guaranteed by the equation in general, but it does help that we know that $$u \in W^{1, 2}$$. This embeds into $$L^{\frac{2n}{n-2}}$$, so if $$n = 2$$ and $$p$$ is anything or $$n > 2$$ and $$p < \frac{n + 2}{n - 2}$$ we succeed.

This kind of argument is "sharp" in the sense that there are examples of solutions to semilinear equations which are not bounded. The most standard is $$u(x) = |x|^{-\frac{2}{p - 1}}$$ being a distributional solution of $$-\Delta u = au^p$$ for some $$a \in \mathbb{R}$$: this $$u$$ is not in $$L^q$$ for $$q \geq \frac{n}{2}(p - 1)$$, and it is not in $$W^{1, 2}$$ unless $$p > \frac{n + 2}{n - 2}$$ (and $$n \geq 3$$). It is not sharp in other ways: it is not sharp at any of the endpoint cases, and the dependence on $$x$$ in the assumptions on $$f(x, u)$$ here is not sharp. Therefore if the question is, in $$n = 2$$ what are the optimal assumptions on $$f$$ to ensure regularity of $$W^{1, 2}$$ solutions, this does not give a complete answer.

• Thank you for this well written answer! However I do have one concern: many authors in the reference I have linked put various assumptions on $f'$ to guarantee a solution of \eqref{1}. I noticed that here, we did not use information about $f'$. Does this mean that regularity is not affected by $f'$? Oct 1, 2021 at 5:09
• Roughly speaking, you always expect $u$ to be two derivatives better than $f$. For $C^{1, \alpha}$ regularity of a bounded $u$, then, boundedness (or less) of $f$ is sufficient, and the argument above shows that this is still valid without a priori knowledge of $u$ being bounded. For higher regularity ($C^{2, \alpha}$, etc.) you will need more information about the smoothness of $f$. Oct 2, 2021 at 4:07
• Assumptions on $f'$ in existence theory are usually of two types: either they are really there to guarantee correct asymptotic behavior of $f$ near infinity (or possibly near $0$) in a strong way, or they are there because the approach towards existence relies on differentiating the equation in some sense (e.g. arguments based on implicit function theorems). Oct 2, 2021 at 4:15
• And might you have any idea why one these assumptions is that $f'$ doesn't hit an eigenvalue of $\Delta$' (or lies strictly between two consecutive eigenvalues, etc...)? And does this assumption in particular have anything to do with regularity? Oct 2, 2021 at 7:09