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Stable solutions to double phase problems involving a nonlocal term

Part of: Elliptic equations and systems Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  23 October 2023

Belgacem Rahal  and
Phuong Le Open the ORCID record for Phuong Le [Opens in a new window]
Belgacem Rahal
Affiliation:
Higher Institute of Computer Science and Management of Kairouan, University of Kairouan, Kairouan, Tunisia (rahhalbelgacem@gmail.com; belgacemrahal.isigk@gmail.com) LR18ES16: Analysis, Geometry and Applications, Faculty of Sciences of Monastir, University of Monastir, Monastir, Tunisia
Phuong Le
Affiliation:
Faculty of Economic Mathematics, University of Economics and Law, Ho Chi Minh City, Vietnam (phuongl@uel.edu.vn) Vietnam National University, Ho Chi Minh City, Vietnam
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Abstract

In this paper, we study weak solutions, possibly unbounded and sign-changing, to the double phase problem

\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*}
where $q\ge p\ge2$, 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.

Keywords

double phase problemsstable solutionsLiouville theoremsHartree nonlinearity

MSC classification

Primary: 35J92: Quasilinear elliptic equations with $p$-Laplacian 35Q40: PDEs in connection with quantum mechanics 35B53: Liouville theorems, Phragmén-Lindelöf theorems 35B35: Stability
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 1119 - 1141
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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Footnotes

The online version of this article has been updated since original publication. A notice detailing the changes has also been published at DOI https://doi.org/10.1017/S0013091523000718

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