Consider the problem
\begin{align*}
(P) \begin{cases}
– \Delta u &= |u|^{q-2}u \ \text{in} \ \Omega,\\
u &= 0 \ \text{on} \ \partial \Omega,
\end{cases}
\end{align*}
where $\Omega \subset \mathbb{R}^N$ is a bounded domain and $2 < q < +\infty$.
I'm trying to find an ordered pair of subsolution and supersolution for this problem in order to find a positive solution for the problem.
A subsolution is a function $\underline{u} \in C^2(\Omega) \cap C(\overline{\Omega})$ such that
\begin{align*}
\begin{cases}
– \Delta \underline{u} &\leq |\underline{u}|^{q-2}\underline{u} \ \text{in} \ \Omega,\\
\underline{u} &\leq 0 \ \text{on} \ \partial \Omega.
\end{cases}
\end{align*}
A supersolution is a function $\overline{u} \in C^2(\Omega) \cap C(\overline{\Omega})$ such that
\begin{align*}
\begin{cases}
– \Delta \overline{u} &\geq |\overline{u}|^{q-2}\overline{u} \ \text{in} \ \Omega,\\
\overline{u} &\geq 0 \ \text{on} \ \partial \Omega.
\end{cases}
\end{align*}
The subsolution and the supersolution are ordered if
\begin{align*}
\begin{cases}
\underline{u} &\leq \overline{u} \ \text{in} \ \Omega,\\
\underline{u} &\leq 0 \leq \overline{u} \ \text{on} \ \partial \Omega.
\end{cases}
\end{align*}
I could show that $\underline{u} := 0$ and $\overline{u} := K w$, where $K > 0$ is a constant small enough and $w$ is a positive solution for the linear problem
\begin{align*}
(P_1) \begin{cases}
– \Delta w &= 1 \ \text{in} \ \Omega,\\
w &= 0 \ \text{on} \ \partial \Omega,
\end{cases}
\end{align*}
is an ordered pair of subsolution and supersolution using comparison principle, but I couldn't show that the solution $u$, which is between $\underline{u}$ and $\overline{u}$, is positive in $\Omega$.
I would like to know if there is some way to show that the problem $(P)$ has a positive solution using the sub-supersolution method.
Thanks in advance!
Best Answer
No it is not possible without further assumptions on either $q$ or $\Omega$. Indeed, if $\Omega$ is star-shaped and $q > \frac{2n}{n-2}$ then there are no non-trivial (by non-trivial we mean $u \not\equiv 0$) solutions. A proof of this can be found in Theorem 3 of Section 9.4.2 in Partial Differential Equations by L. Evans.