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Stable solutions to double phase problems involving a nonlocal term
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 66 / Issue 4 / November 2023
- Published online by Cambridge University Press:
- 23 October 2023, pp. 1119-1141
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In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem
where
\begin{equation*}-\text{div} (|\nabla u|^{p-2} \nabla u + w(x)|\nabla u|^{q-2} \nabla u) = \left(\frac{1}{|x|^{N-\mu}}*f|u|^r\right) f(x)|u|^{r-2}u \quad\text{in}\ \mathbb{R}^N,\end{equation*}
$q\ge p\ge2$, r > q, 
$0 \lt \mu \lt N$ and 
$w,f \in L^1_{\rm loc}(\mathbb{R}^N)$ are two non-negative functions such that 
$w(x) \le C_1|x|^a$ and 
$f(x) \ge C_2|x|^b$ for all 
$|x| \gt R_0$, where 
$R_0,C_1,C_2 \gt 0$ and 
$a,b\in\mathbb{R}$. Under some appropriate assumptions on p, q, r, µ, a, b and N, we prove various Liouville-type theorems for weak solutions which are stable or stable outside a compact set of 
$\mathbb{R}^N$. First, we establish the standard integral estimates via stability property to derive the non-existence results for stable weak solutions. Then, by means of the Pohožaev identity, we deduce the Liouville-type theorem for weak solutions which are stable outside a compact set.
