# Strong maximum principle for a PDE with coefficient in $L^1$

Let $$U$$ be a bounded open subset of $$\mathbb{R}^n$$ with smooth boundary, $$n \geq 3$$. Set $$N = \frac{2n}{n-2}$$. I am interested in the following equation: $$-\Delta \phi + R \phi + \phi^{N-1} = 0$$ with boundary condition $$\phi \equiv 1$$ on $$\partial U$$. I also assume that $$R \in L^1(U)$$, $$R \geq -1$$.

It turns out that we can construct $$\phi \in W^{1,2}(U) \cap L^\infty(U)$$ hwich is a weak solution to this equation as follows.

First a naive remark. If $$R$$ was regular, we see that $$1$$ is a supersolution while $$0$$ is a (sub) solution so we expect $$0 \leq \phi \leq 1$$.

We introduce the functional $$I(\phi) = \int_U \left[\frac{1}{2}\left(|d\phi|^2 + R \phi^2\right) + \frac{1}{N} |\phi|^N\right] dx.$$ This functional is ill-defined over $$W^{1,2}(U)$$ but well-defined over the closed subset $$C$$ consisting of functions that are between $$-2$$ and $$2$$ ($$C$$ is closed for the weak topology as it can be described as the set of functions $$\phi$$ such that $$-2 \mu(A) \leq \int_A \phi dx \leq 2 \mu(A)$$ for all measurable $$A \subset U$$).

By classical arguments $$I$$ is sequentially lower semicontinuous on $$C$$ and, hence, admits a minimum $$\phi_0$$. As $$I(\max\{\phi_0, 0\}) \leq I(\phi_0)$$ and $$I(\min\{\phi_0, 1\}) \leq I(\phi_0)$$, we can assume that $$0 \leq \phi_0 \leq 1$$.

As we have allowed ourselves some wiggle room in the definition of $$C$$, we have, for each $$\psi \in W^{1, 2}_0(U) \cap L^\infty(U)$$, that $$\phi_0 + \lambda \psi \in C$$ for small enough $$\lambda$$ and, hence, "differentiating" the inequality $$I(\phi_0 + \lambda \psi) \geq I(\phi_0)$$, we get that, for all such $$\psi$$, $$0 = \int_U \left(\langle d\phi_0, d\psi\rangle + R \phi_0 \psi + \phi^{N-1} \psi\right) dx,$$ i.e. $$\phi_0$$ is a weak solution to our problem.

Is $$\phi_0$$ bounded from below by a positive constant?

With a more regular $$R$$, that would follow from Harnack's inequality but I could not find a proof with regularity as low as what I presented.

For the interested reader, the context in which this problem appears is the study of the mass of asymptotically hyperbolic manifolds (see e.g. https://arxiv.org/abs/math/0110035).